From d98d2bcab0dae803dfa625ebf3509f3218f5bb6d Mon Sep 17 00:00:00 2001
From: Balasubramanian Narasimhan <b.naras@gmail.com>
Date: Tue, 4 Aug 2026 17:29:18 -0700
Subject: [PATCH 1/3] Fix Rd math markup per CRAN; bump version to 1.2-2

---
 DESCRIPTION                |  4 ++--
 NEWS.md                    |  8 ++++++++
 R/deconv.R                 |  6 +++---
 R/deconvolveR-package.R    |  5 +----
 man/deconv.Rd              |  6 +++---
 man/deconvolveR-package.Rd | 19 +++++++++++++++++++
 6 files changed, 36 insertions(+), 12 deletions(-)

diff --git a/DESCRIPTION b/DESCRIPTION
index a91506a..42d55c6 100644
--- a/DESCRIPTION
+++ b/DESCRIPTION
@@ -1,6 +1,6 @@
 Package: deconvolveR
 Title: Empirical Bayes Estimation Strategies
-Version: 1.2-1
+Version: 1.2-2
 VignetteBuilder: knitr
 Suggests: cowplot, ggplot2, knitr, rmarkdown
 Authors@R: c(person("Bradley", "Efron", role=c("aut"),
@@ -23,4 +23,4 @@ LazyData: true
 Imports:
     splines,
     stats
-RoxygenNote: 7.1.0
+Config/roxygen2/version: 8.0.0
diff --git a/NEWS.md b/NEWS.md
index 6780ee9..ff6fb98 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -1,3 +1,11 @@
+# deconvolveR 1.2-2
+
+- Fixed documentation markup flagged by CRAN (`checkRd` "Lost braces"
+  NOTE in `deconv.Rd`)
+- Switched CI from Travis to GitHub Actions; pkgdown site is now
+  built and deployed via GitHub Actions
+- Added a package logo and updated badges
+
 # deconvolveR 1.2-1
 
 - Fixed URL per CRAN
diff --git a/R/deconv.R b/R/deconv.R
index e928564..2ba3fc3 100644
--- a/R/deconv.R
+++ b/R/deconv.R
@@ -45,9 +45,9 @@
 #'     where it was referred to as \eqn{R(\alpha)}} \item{cov}{the
 #'     covariance matrix for the mle} \item{cov.g}{the covariance
 #'     matrix for the \eqn{g}} \item{stats}{an m by 6 or 7 matrix
-#'     containing columns for \eqn{theta}, \eqn{g}, \eqn{\tilde{g}}
+#'     containing columns for \eqn{\theta}, \eqn{g}, \eqn{\tilde{g}}
 #'     which is \eqn{g} with thinning correction applied and named
-#'     \code{tg}, std. error of \eqn{g}, \eqn{G} (the cdf of {g}),
+#'     \code{tg}, std. error of \eqn{g}, \eqn{G} (the cdf of \eqn{g}),
 #'     std. error of \eqn{G}, and the bias of \eqn{g}}
 #'     \item{loglik}{the negative log-likelihood function for the data
 #'     taking a \eqn{p}-vector argument} \item{statsFunction}{a
@@ -56,7 +56,7 @@
 #' @section Details:
 #' The data \code{X} is always required with two exceptions. In the Poisson case,
 #' \code{y} alone may be specified and \code{X} omitted, in which case the sample space of
-#' the observations $\eqn{X}$ is assumed to be 1, 2, .., \code{length(y)}. The second exception is
+#' the observations \eqn{X} is assumed to be 1, 2, .., \code{length(y)}. The second exception is
 #' for experimentation with other exponential families besides the three implemented here:
 #' \code{y}, \code{P} and \code{Q} can be specified together.
 #'
diff --git a/R/deconvolveR-package.R b/R/deconvolveR-package.R
index 4c73ca0..0377cf1 100644
--- a/R/deconvolveR-package.R
+++ b/R/deconvolveR-package.R
@@ -5,9 +5,6 @@
 #' paper reference below and this package includes a vignette
 #' containing a number of examples.
 #'
-#' @docType package
-#' @name deconvolveR-package
-#'
 #' @references Bradley Efron. Empirical Bayes Deconvolution
 #'     Estimates. Biometrika 103(1), 1-20, ISSN
 #'     0006-3444. doi:10.1093/biomet/asv068.
@@ -19,7 +16,7 @@
 #'     Efron. deconvolveR: A G-Modeling Program for Deconvolution and
 #'     Empirical Bayes Estimation. doi:10.18637/jss.v094.i11
 #'
-NULL
+"_PACKAGE"
 
 #' Shakespeare word counts in the entire canon: 14,376 distinct words
 #' appeared exactly once, 4343 words appeared twice etc.
diff --git a/man/deconv.Rd b/man/deconv.Rd
index 70cbb57..0f256f0 100644
--- a/man/deconv.Rd
+++ b/man/deconv.Rd
@@ -80,9 +80,9 @@ a list of 9 items consisting of \item{mle}{the maximum
     where it was referred to as \eqn{R(\alpha)}} \item{cov}{the
     covariance matrix for the mle} \item{cov.g}{the covariance
     matrix for the \eqn{g}} \item{stats}{an m by 6 or 7 matrix
-    containing columns for \eqn{theta}, \eqn{g}, \eqn{\tilde{g}}
+    containing columns for \eqn{\theta}, \eqn{g}, \eqn{\tilde{g}}
     which is \eqn{g} with thinning correction applied and named
-    \code{tg}, std. error of \eqn{g}, \eqn{G} (the cdf of {g}),
+    \code{tg}, std. error of \eqn{g}, \eqn{G} (the cdf of \eqn{g}),
     std. error of \eqn{G}, and the bias of \eqn{g}}
     \item{loglik}{the negative log-likelihood function for the data
     taking a \eqn{p}-vector argument} \item{statsFunction}{a
@@ -95,7 +95,7 @@ A function to compute Empirical Bayes estimates using deconvolution
 
 The data \code{X} is always required with two exceptions. In the Poisson case,
 \code{y} alone may be specified and \code{X} omitted, in which case the sample space of
-the observations $\eqn{X}$ is assumed to be 1, 2, .., \code{length(y)}. The second exception is
+the observations \eqn{X} is assumed to be 1, 2, .., \code{length(y)}. The second exception is
 for experimentation with other exponential families besides the three implemented here:
 \code{y}, \code{P} and \code{Q} can be specified together.
 
diff --git a/man/deconvolveR-package.Rd b/man/deconvolveR-package.Rd
index 9625b7e..2728b43 100644
--- a/man/deconvolveR-package.Rd
+++ b/man/deconvolveR-package.Rd
@@ -2,6 +2,7 @@
 % Please edit documentation in R/deconvolveR-package.R
 \docType{package}
 \name{deconvolveR-package}
+\alias{deconvolveR}
 \alias{deconvolveR-package}
 \title{R package for Empirical Bayes \eqn{g}-modeling using exponential families.}
 \description{
@@ -24,3 +25,21 @@ Balasubramanian Narasimhan and Bradley
     Efron. deconvolveR: A G-Modeling Program for Deconvolution and
     Empirical Bayes Estimation. doi:10.18637/jss.v094.i11
 }
+\seealso{
+Useful links:
+\itemize{
+  \item \url{https://bnaras.github.io/deconvolveR/}
+  \item Report bugs at \url{https://github.com/bnaras/deconvolveR/issues}
+}
+
+}
+\author{
+\strong{Maintainer}: Balasubramanian Narasimhan \email{naras@stat.Stanford.EDU}
+
+Authors:
+\itemize{
+  \item Balasubramanian Narasimhan \email{naras@stat.Stanford.EDU}
+  \item Bradley Efron \email{brad@stat.stanford.edu}
+}
+
+}

From b6acd9250a3473bf95c46e1bb3debb28974867fd Mon Sep 17 00:00:00 2001
From: Balasubramanian Narasimhan <b.naras@gmail.com>
Date: Tue, 4 Aug 2026 17:29:18 -0700
Subject: [PATCH 2/3] Replace Travis CI with GitHub Actions; deploy pkgdown
 site to gh-pages

---
 .Rbuildignore                                 |    3 +-
 .github/.gitignore                            |    1 +
 .github/workflows/R-CMD-check.yaml            |   50 +
 .github/workflows/pkgdown.yaml                |   47 +
 .gitignore                                    |    2 +
 .travis.yml                                   |   11 -
 README.md                                     |   14 +-
 docs/404.html                                 |  160 ---
 docs/articles/deconvolution.html              | 1250 -----------------
 .../figure-html/unnamed-chunk-11-1.png        |  Bin 74273 -> 0 bytes
 .../figure-html/unnamed-chunk-13-1.png        |  Bin 86474 -> 0 bytes
 .../figure-html/unnamed-chunk-14-1.png        |  Bin 176108 -> 0 bytes
 .../figure-html/unnamed-chunk-15-1.png        |  Bin 101419 -> 0 bytes
 .../figure-html/unnamed-chunk-16-1.png        |  Bin 62027 -> 0 bytes
 .../figure-html/unnamed-chunk-19-1.png        |  Bin 61117 -> 0 bytes
 .../figure-html/unnamed-chunk-22-1.png        |  Bin 81480 -> 0 bytes
 .../figure-html/unnamed-chunk-27-1.png        |  Bin 171866 -> 0 bytes
 .../figure-html/unnamed-chunk-28-1.png        |  Bin 72808 -> 0 bytes
 .../figure-html/unnamed-chunk-30-1.png        |  Bin 408254 -> 0 bytes
 .../figure-html/unnamed-chunk-32-1.png        |  Bin 67298 -> 0 bytes
 .../figure-html/unnamed-chunk-36-1.png        |  Bin 96808 -> 0 bytes
 .../figure-html/unnamed-chunk-7-1.png         |  Bin 121632 -> 0 bytes
 .../header-attrs-2.3/header-attrs.js          |   12 -
 docs/articles/index.html                      |  159 ---
 docs/authors.html                             |  184 ---
 docs/bootstrap-toc.css                        |   60 -
 docs/bootstrap-toc.js                         |  159 ---
 docs/docsearch.css                            |  148 --
 docs/docsearch.js                             |   85 --
 docs/index.html                               |  158 ---
 docs/link.svg                                 |   12 -
 docs/news/index.html                          |  191 ---
 docs/pkgdown.css                              |  367 -----
 docs/pkgdown.js                               |  108 --
 docs/pkgdown.yml                              |    7 -
 docs/reference/bardWordCount.html             |  176 ---
 docs/reference/deconv-1.png                   |  Bin 90940 -> 0 bytes
 docs/reference/deconv.html                    |  359 -----
 docs/reference/deconvolveR-package.html       |  183 ---
 docs/reference/deconvolveR.html               |  183 ---
 docs/reference/disjointTheta.html             |  166 ---
 docs/reference/index.html                     |  212 ---
 docs/reference/surg.html                      |  181 ---
 43 files changed, 111 insertions(+), 4537 deletions(-)
 create mode 100644 .github/.gitignore
 create mode 100644 .github/workflows/R-CMD-check.yaml
 create mode 100644 .github/workflows/pkgdown.yaml
 delete mode 100644 .travis.yml
 delete mode 100644 docs/404.html
 delete mode 100644 docs/articles/deconvolution.html
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-11-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-13-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-14-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-15-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-16-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-19-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-22-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-27-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-28-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-30-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-32-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-36-1.png
 delete mode 100644 docs/articles/deconvolution_files/figure-html/unnamed-chunk-7-1.png
 delete mode 100644 docs/articles/deconvolution_files/header-attrs-2.3/header-attrs.js
 delete mode 100644 docs/articles/index.html
 delete mode 100644 docs/authors.html
 delete mode 100644 docs/bootstrap-toc.css
 delete mode 100644 docs/bootstrap-toc.js
 delete mode 100644 docs/docsearch.css
 delete mode 100644 docs/docsearch.js
 delete mode 100644 docs/index.html
 delete mode 100644 docs/link.svg
 delete mode 100644 docs/news/index.html
 delete mode 100644 docs/pkgdown.css
 delete mode 100644 docs/pkgdown.js
 delete mode 100644 docs/pkgdown.yml
 delete mode 100644 docs/reference/bardWordCount.html
 delete mode 100644 docs/reference/deconv-1.png
 delete mode 100644 docs/reference/deconv.html
 delete mode 100644 docs/reference/deconvolveR-package.html
 delete mode 100644 docs/reference/deconvolveR.html
 delete mode 100644 docs/reference/disjointTheta.html
 delete mode 100644 docs/reference/index.html
 delete mode 100644 docs/reference/surg.html

diff --git a/.Rbuildignore b/.Rbuildignore
index 31f1dde..bd28d11 100644
--- a/.Rbuildignore
+++ b/.Rbuildignore
@@ -1,7 +1,6 @@
 ^.*\.Rproj$
 ^\.Rhistory$
 ^\.Rproj\.user$
-^\.travis\.yml$
 ^docs$
 ^_pkgdown\.yml$
 ^README\.Rmd$
@@ -11,3 +10,5 @@
 ^LICENSE\.md$
 ^NOTES\.md$
 ^CRAN-RELEASE$
+^\.github$
+^tools$
diff --git a/.github/.gitignore b/.github/.gitignore
new file mode 100644
index 0000000..2d19fc7
--- /dev/null
+++ b/.github/.gitignore
@@ -0,0 +1 @@
+*.html
diff --git a/.github/workflows/R-CMD-check.yaml b/.github/workflows/R-CMD-check.yaml
new file mode 100644
index 0000000..73bfb21
--- /dev/null
+++ b/.github/workflows/R-CMD-check.yaml
@@ -0,0 +1,50 @@
+# Workflow derived from https://github.com/r-lib/actions/tree/v2/examples
+# Need help debugging build failures? Start at https://github.com/r-lib/actions#where-to-find-help
+on:
+  push:
+    branches: [main, master]
+  pull_request:
+
+name: R-CMD-check.yaml
+
+permissions: read-all
+
+jobs:
+  R-CMD-check:
+    runs-on: ${{ matrix.config.os }}
+
+    name: ${{ matrix.config.os }} (${{ matrix.config.r }})
+
+    strategy:
+      fail-fast: false
+      matrix:
+        config:
+          - {os: macos-latest,   r: 'release'}
+          - {os: windows-latest, r: 'release'}
+          - {os: ubuntu-latest,   r: 'devel', http-user-agent: 'release'}
+          - {os: ubuntu-latest,   r: 'release'}
+          - {os: ubuntu-latest,   r: 'oldrel-1'}
+
+    env:
+      GITHUB_PAT: ${{ secrets.GITHUB_TOKEN }}
+      R_KEEP_PKG_SOURCE: yes
+
+    steps:
+      - uses: actions/checkout@v6
+
+      - uses: r-lib/actions/setup-pandoc@v2
+
+      - uses: r-lib/actions/setup-r@v2
+        with:
+          r-version: ${{ matrix.config.r }}
+          http-user-agent: ${{ matrix.config.http-user-agent }}
+
+      - uses: r-lib/actions/setup-r-dependencies@v2
+        with:
+          extra-packages: any::rcmdcheck
+          needs: check
+
+      - uses: r-lib/actions/check-r-package@v2
+        with:
+          upload-snapshots: true
+          build_args: 'c("--no-manual","--compact-vignettes=gs+qpdf")'
diff --git a/.github/workflows/pkgdown.yaml b/.github/workflows/pkgdown.yaml
new file mode 100644
index 0000000..16c6c8b
--- /dev/null
+++ b/.github/workflows/pkgdown.yaml
@@ -0,0 +1,47 @@
+# Workflow derived from https://github.com/r-lib/actions/tree/v2/examples
+# Need help debugging build failures? Start at https://github.com/r-lib/actions#where-to-find-help
+on:
+  push:
+    branches: [main, master]
+  pull_request:
+  release:
+    types: [published]
+  workflow_dispatch:
+
+name: pkgdown.yaml
+
+permissions: read-all
+
+jobs:
+  pkgdown:
+    runs-on: ubuntu-latest
+    # Only restrict concurrency for non-PR jobs
+    concurrency:
+      group: pkgdown-${{ github.event_name != 'pull_request' || github.run_id }}
+    env:
+      GITHUB_PAT: ${{ secrets.GITHUB_TOKEN }}
+    permissions:
+      contents: write
+    steps:
+      - uses: actions/checkout@v6
+
+      - uses: r-lib/actions/setup-pandoc@v2
+
+      - uses: r-lib/actions/setup-r@v2
+
+      - uses: r-lib/actions/setup-r-dependencies@v2
+        with:
+          extra-packages: any::pkgdown, local::.
+          needs: website
+
+      - name: Build site
+        run: pkgdown::build_site_github_pages(new_process = FALSE, install = FALSE)
+        shell: Rscript {0}
+
+      - name: Deploy to GitHub pages 🚀
+        if: github.event_name != 'pull_request'
+        uses: JamesIves/github-pages-deploy-action@d92aa235d04922e8f08b40ce78cc5442fcfbfa2f # v4.8.0
+        with:
+          clean: false
+          branch: gh-pages
+          folder: docs
diff --git a/.gitignore b/.gitignore
index 5a1121e..c5a3e76 100644
--- a/.gitignore
+++ b/.gitignore
@@ -2,3 +2,5 @@
 .Rproj.user
 .RData
 .Rhistory
+*~
+docs/
diff --git a/.travis.yml b/.travis.yml
deleted file mode 100644
index 52bc8ae..0000000
--- a/.travis.yml
+++ /dev/null
@@ -1,11 +0,0 @@
-language: r
-
-r:
-  - release
-  - devel
-cache: packages
-
-sudo: false
-
-notifications:
-  email: false
diff --git a/README.md b/README.md
index ecf0fad..784761f 100644
--- a/README.md
+++ b/README.md
@@ -1,9 +1,13 @@
-Empirical Bayes Deconvolution
-=============================
+# deconvolveR <img src="man/figures/logo.png" align="right" height="139" alt="deconvolveR hex logo" />
 
-[![Travis-CI Build
-Status](https://travis-ci.org/bnaras/deconvolveR.svg?branch=master)](https://travis-ci.org/bnaras/deconvolveR)
-[![CRAN_Status_Badge](https://www.r-pkg.org/badges/version/deconvolveR)](https://cran.r-project.org/package=deconvolveR)
+<!-- badges: start -->
+[![R-CMD-check](https://github.com/bnaras/deconvolveR/actions/workflows/R-CMD-check.yaml/badge.svg)](https://github.com/bnaras/deconvolveR/actions/workflows/R-CMD-check.yaml)
+[![CRAN status](https://www.r-pkg.org/badges/version/deconvolveR)](https://cran.r-project.org/package=deconvolveR)
+[![Monthly downloads](https://cranlogs.r-pkg.org/badges/deconvolveR)](https://cran.r-project.org/package=deconvolveR)
+[![Total downloads](https://cranlogs.r-pkg.org/badges/grand-total/deconvolveR)](https://cran.r-project.org/package=deconvolveR)
+<!-- badges: end -->
+
+## Empirical Bayes Deconvolution
 
 An unknown prior density $g(\theta)$ has yielded (unobservable)
 $\Theta_1, \Theta_2,\ldots,\Theta_N$, and each $\Theta_i$ produces an
diff --git a/docs/404.html b/docs/404.html
deleted file mode 100644
index d65c6a2..0000000
--- a/docs/404.html
+++ /dev/null
@@ -1,160 +0,0 @@
-<!-- Generated by pkgdown: do not edit by hand -->
-<!DOCTYPE html>
-<html lang="en">
-  <head>
-  <meta charset="utf-8">
-<meta http-equiv="X-UA-Compatible" content="IE=edge">
-<meta name="viewport" content="width=device-width, initial-scale=1.0">
-
-<title>Page not found (404) • deconvolveR</title>
-
-
-<!-- jquery -->
-<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.4.1/jquery.min.js" integrity="sha256-CSXorXvZcTkaix6Yvo6HppcZGetbYMGWSFlBw8HfCJo=" crossorigin="anonymous"></script>
-<!-- Bootstrap -->
-
-<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/css/bootstrap.min.css" integrity="sha256-bZLfwXAP04zRMK2BjiO8iu9pf4FbLqX6zitd+tIvLhE=" crossorigin="anonymous" />
-
-<script src="https://cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.4.1/js/bootstrap.min.js" integrity="sha256-nuL8/2cJ5NDSSwnKD8VqreErSWHtnEP9E7AySL+1ev4=" crossorigin="anonymous"></script>
-
-<!-- bootstrap-toc -->
-<link rel="stylesheet" href="bootstrap-toc.css">
-<script src="bootstrap-toc.js"></script>
-
-<!-- Font Awesome icons -->
-<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/all.min.css" integrity="sha256-mmgLkCYLUQbXn0B1SRqzHar6dCnv9oZFPEC1g1cwlkk=" crossorigin="anonymous" />
-<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.12.1/css/v4-shims.min.css" integrity="sha256-wZjR52fzng1pJHwx4aV2AO3yyTOXrcDW7jBpJtTwVxw=" crossorigin="anonymous" />
-
-<!-- clipboard.js -->
-<script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js" integrity="sha256-inc5kl9MA1hkeYUt+EC3BhlIgyp/2jDIyBLS6k3UxPI=" crossorigin="anonymous"></script>
-
-<!-- headroom.js -->
-<script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/headroom.min.js" integrity="sha256-AsUX4SJE1+yuDu5+mAVzJbuYNPHj/WroHuZ8Ir/CkE0=" crossorigin="anonymous"></script>
-<script src="https://cdnjs.cloudflare.com/ajax/libs/headroom/0.11.0/jQuery.headroom.min.js" integrity="sha256-ZX/yNShbjqsohH1k95liqY9Gd8uOiE1S4vZc+9KQ1K4=" crossorigin="anonymous"></script>
-
-<!-- pkgdown -->
-<link href="pkgdown.css" rel="stylesheet">
-<script src="pkgdown.js"></script>
-
-
-
-
-<meta property="og:title" content="Page not found (404)" />
-
-
-
-
-<!-- mathjax -->
-<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js" integrity="sha256-nvJJv9wWKEm88qvoQl9ekL2J+k/RWIsaSScxxlsrv8k=" crossorigin="anonymous"></script>
-<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/config/TeX-AMS-MML_HTMLorMML.js" integrity="sha256-84DKXVJXs0/F8OTMzX4UR909+jtl4G7SPypPavF+GfA=" crossorigin="anonymous"></script>
-
-<!--[if lt IE 9]>
-<script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script>
-<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
-<![endif]-->
-
-
-
-  </head>
-
-  <body data-spy="scroll" data-target="#toc">
-    <div class="container template-title-body">
-      <header>
-      <div class="navbar navbar-default navbar-fixed-top" role="navigation">
-  <div class="container">
-    <div class="navbar-header">
-      <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false">
-        <span class="sr-only">Toggle navigation</span>
-        <span class="icon-bar"></span>
-        <span class="icon-bar"></span>
-        <span class="icon-bar"></span>
-      </button>
-      <span class="navbar-brand">
-        <a class="navbar-link" href="index.html">deconvolveR</a>
-        <span class="version label label-default" data-toggle="tooltip" data-placement="bottom" title="Released version">1.2-1</span>
-      </span>
-    </div>
-
-    <div id="navbar" class="navbar-collapse collapse">
-      <ul class="nav navbar-nav">
-        <li>
-  <a href="index.html">
-    <span class="fas fa fas fa-home fa-lg"></span>
-     
-  </a>
-</li>
-<li>
-  <a href="reference/index.html">Reference</a>
-</li>
-<li class="dropdown">
-  <a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-expanded="false">
-    Articles
-     
-    <span class="caret"></span>
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-      <h1 data-toc-skip>Empirical Bayes Deconvolution</h1>
-                        <h4 class="author">Balasubramanian Narasimhan and Bradley Efron</h4>
-            
-            <h4 class="date">2020-08-29</h4>
-      
-      <small class="dont-index">Source: <a href="https://github.com/bnaras/deconvolveR/blob/master/vignettes/deconvolution.Rmd"><code>vignettes/deconvolution.Rmd</code></a></small>
-      <div class="hidden name"><code>deconvolution.Rmd</code></div>
-
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-    
-<div id="a-simulation-example" class="section level2">
-<h2 class="hasAnchor">
-<a href="#a-simulation-example" class="anchor"></a>A simulation example</h2>
-<p>We start with a simulated Poisson example where the <span class="math inline">\(\Theta_i\)</span> are drawn from a chi-squared density with 10 degrees of freedom and the <span class="math inline">\(X_i|\Theta_i\)</span> are Poisson with expectation <span class="math inline">\(\Theta_i:\)</span></p>
-<p><span class="math display">\[
-\Theta_i \sim \chi^2_{10} \mbox{ and } X_i|\Theta_i \sim \mbox{Poisson}(\Theta_i)
-\]</span></p>
-<p>The <span class="math inline">\(\Theta_i\)</span> for this setting, with <code>N = 1000</code> observations can be generated as follows.</p>
-<div class="sourceCode" id="cb1"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/Random.html">set.seed</a></span>(<span class="fl">238923</span>) <span class="co">## for reproducibility</span>
-<span class="no">N</span> <span class="kw">&lt;-</span> <span class="fl">1000</span>
-<span class="no">Theta</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Chisquare.html">rchisq</a></span>(<span class="no">N</span>,  <span class="kw">df</span> <span class="kw">=</span> <span class="fl">10</span>)</pre></body></html></div>
-<p>Next, the <span class="math inline">\(X_i|\Theta_i\)</span>, for each of <code>nSIM = 1000</code> simulations can be generated as below.</p>
-<div class="sourceCode" id="cb2"><html><body><pre class="r"><span class="no">nSIM</span> <span class="kw">&lt;-</span> <span class="fl">1000</span>
-<span class="no">data</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq_len</a></span>(<span class="no">nSIM</span>), <span class="kw">function</span>(<span class="no">x</span>) <span class="fu"><a href="https://rdrr.io/r/stats/Poisson.html">rpois</a></span>(<span class="kw">n</span> <span class="kw">=</span> <span class="no">N</span>, <span class="kw">lambda</span> <span class="kw">=</span> <span class="no">Theta</span>))</pre></body></html></div>
-<p>We take the discrete set <span class="math inline">\(\mathcal{T}=(1, 2, \ldots, 32)\)</span> as the <span class="math inline">\(\Theta\)</span>-space and apply the <code>deconv</code> function in the package <code>deconvolveR</code> to estimate <span class="math inline">\(g(\theta).\)</span></p>
-<div class="sourceCode" id="cb3"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/library.html">library</a></span>(<span class="no">deconvolveR</span>)
-<span class="no">tau</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="fl">1</span>, <span class="fl">32</span>)
-<span class="no">results</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">data</span>, <span class="fl">2</span>,
-                 <span class="kw">function</span>(<span class="no">x</span>) <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">x</span>, <span class="kw">ignoreZero</span> <span class="kw">=</span> <span class="fl">FALSE</span>,
-                                    <span class="kw">c0</span> <span class="kw">=</span> <span class="fl">1</span>))</pre></body></html></div>
-<p>The default setting for <code>deconv</code> uses the <code>Poisson</code> family and a natural cubic spline basis of degree 5 as <span class="math inline">\(Q.\)</span> The regularization parameter for this example (<code>c0</code>) is set to 1. The <code>ignoreZero</code> parameter indicates that this dataset contains zero counts, i.e., there zeros have not been truncated. (In the Shakespeare example below, the counts are of words seen in the canon, and so there is a natural truncation at zero.)</p>
-<p>Some warnings are emitted by the <code>nlm</code> routine used for optimization, but they are mostly inconsequential.</p>
-<p>Since <code>deconv</code> works on a sample at a time, the result above is a list of lists from which various statistics can be extracted. Below, we construct a table of values for various values of <span class="math inline">\(\Theta\)</span>.</p>
-<div class="sourceCode" id="cb4"><html><body><pre class="r"><span class="no">g</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">results</span>, <span class="kw">function</span>(<span class="no">x</span>) <span class="no">x</span>$<span class="no">stats</span>[, <span class="st">"g"</span>])
-<span class="no">mean</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">g</span>, <span class="fl">1</span>, <span class="no">mean</span>)
-<span class="no">SE.g</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">results</span>, <span class="kw">function</span>(<span class="no">x</span>) <span class="no">x</span>$<span class="no">stats</span>[, <span class="st">"SE.g"</span>])
-<span class="no">sd</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">SE.g</span>, <span class="fl">1</span>, <span class="no">mean</span>)
-<span class="no">Bias.g</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">results</span>, <span class="kw">function</span>(<span class="no">x</span>) <span class="no">x</span>$<span class="no">stats</span>[, <span class="st">"Bias.g"</span>])
-<span class="no">bias</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">Bias.g</span>, <span class="fl">1</span>, <span class="no">mean</span>)
-<span class="no">gTheta</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Chisquare.html">pchisq</a></span>(<span class="no">tau</span>, <span class="kw">df</span> <span class="kw">=</span> <span class="fl">10</span>) - <span class="fu"><a href="https://rdrr.io/r/stats/Chisquare.html">pchisq</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">0</span>, <span class="no">tau</span>[-<span class="fu"><a href="https://rdrr.io/r/base/length.html">length</a></span>(<span class="no">tau</span>)]), <span class="kw">df</span> <span class="kw">=</span> <span class="fl">10</span>)
-<span class="no">gTheta</span> <span class="kw">&lt;-</span> <span class="no">gTheta</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">gTheta</span>)
-<span class="no">simData</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(<span class="kw">theta</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">gTheta</span> <span class="kw">=</span> <span class="no">gTheta</span>,
-                      <span class="kw">Mean</span> <span class="kw">=</span> <span class="no">mean</span>, <span class="kw">StdDev</span> <span class="kw">=</span> <span class="no">sd</span>, <span class="kw">Bias</span> <span class="kw">=</span> <span class="no">bias</span>,
-                      <span class="kw">CoefVar</span> <span class="kw">=</span> <span class="no">sd</span> / <span class="no">mean</span>)
-<span class="no">table1</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/transform.html">transform</a></span>(<span class="no">simData</span>,
-                    <span class="kw">gTheta</span> <span class="kw">=</span> <span class="fl">100</span> * <span class="no">gTheta</span>,
-                    <span class="kw">Mean</span> <span class="kw">=</span> <span class="fl">100</span> * <span class="no">Mean</span>,
-                    <span class="kw">StdDev</span> <span class="kw">=</span> <span class="fl">100</span> * <span class="no">StdDev</span>,
-                    <span class="kw">Bias</span> <span class="kw">=</span> <span class="fl">100</span> * <span class="no">Bias</span>)</pre></body></html></div>
-<p>The table below summarizes the results for some chosen values of <span class="math inline">\(\theta .\)</span></p>
-<div class="sourceCode" id="cb5"><html><body><pre class="r"><span class="kw pkg">knitr</span><span class="kw ns">::</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html">kable</a></span>(<span class="no">table1</span>[<span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">5</span>, <span class="fl">10</span>, <span class="fl">15</span>, <span class="fl">20</span>, <span class="fl">25</span>), ], <span class="kw">row.names</span><span class="kw">=</span><span class="fl">FALSE</span>)</pre></body></html></div>
-<table class="table">
-<thead><tr class="header">
-<th align="right">theta</th>
-<th align="right">gTheta</th>
-<th align="right">Mean</th>
-<th align="right">StdDev</th>
-<th align="right">Bias</th>
-<th align="right">CoefVar</th>
-</tr></thead>
-<tbody>
-<tr class="odd">
-<td align="right">5</td>
-<td align="right">5.6191465</td>
-<td align="right">5.4416722</td>
-<td align="right">0.3635378</td>
-<td align="right">-0.1235865</td>
-<td align="right">0.0668063</td>
-</tr>
-<tr class="even">
-<td align="right">10</td>
-<td align="right">9.1646990</td>
-<td align="right">9.5316279</td>
-<td align="right">0.4917673</td>
-<td align="right">0.2588187</td>
-<td align="right">0.0515932</td>
-</tr>
-<tr class="odd">
-<td align="right">15</td>
-<td align="right">4.0946148</td>
-<td align="right">3.3421426</td>
-<td align="right">0.3119862</td>
-<td align="right">-0.0683187</td>
-<td align="right">0.0933492</td>
-</tr>
-<tr class="even">
-<td align="right">20</td>
-<td align="right">1.1014405</td>
-<td align="right">0.9810001</td>
-<td align="right">0.2246999</td>
-<td align="right">-0.1225092</td>
-<td align="right">0.2290519</td>
-</tr>
-<tr class="odd">
-<td align="right">25</td>
-<td align="right">0.2255788</td>
-<td align="right">0.1496337</td>
-<td align="right">0.0677919</td>
-<td align="right">0.0602422</td>
-<td align="right">0.4530522</td>
-</tr>
-</tbody>
-</table>
-<p>Although, the coefficient of variation of <span class="math inline">\(\hat{g}(\theta)\)</span> is still large, the <span class="math inline">\(g(\theta)\)</span> estimates are reasonable.</p>
-<p>We can compare the empirical standard deviations and biases of <span class="math inline">\(g(\hat{\alpha})\)</span> with the approximation given by the formulas in the paper.</p>
-<div class="sourceCode" id="cb6"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/library.html">library</a></span>(<span class="no">ggplot2</span>)
-<span class="fu"><a href="https://rdrr.io/r/base/library.html">library</a></span>(<span class="no">cowplot</span>)
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/theme_get.html">theme_set</a></span>(<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/theme_get.html">theme_get</a></span>() +
-          <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/theme.html">theme</a></span>(<span class="kw">panel.grid.major</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/element.html">element_line</a></span>(<span class="kw">colour</span> <span class="kw">=</span> <span class="st">"gray90"</span>,
-                                                <span class="kw">size</span> <span class="kw">=</span> <span class="fl">0.2</span>),
-                <span class="kw">panel.grid.minor</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/element.html">element_line</a></span>(<span class="kw">colour</span> <span class="kw">=</span> <span class="st">"gray98"</span>,
-                                                <span class="kw">size</span> <span class="kw">=</span> <span class="fl">0.5</span>)))
-<span class="no">p1</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/as.data.frame.html">as.data.frame</a></span>(<span class="no">results</span><span class="kw">[[</span><span class="fl">1</span>]]$<span class="no">stats</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">SE.g</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"black"</span>, <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"solid"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">simData</span>$<span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">simData</span>$<span class="no">StdDev</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"red"</span>, <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"dashed"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Std. Dev"</span>)
-
-<span class="no">p2</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/as.data.frame.html">as.data.frame</a></span>(<span class="no">results</span><span class="kw">[[</span><span class="fl">1</span>]]$<span class="no">stats</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">Bias.g</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"black"</span>, <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"solid"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">simData</span>$<span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">simData</span>$<span class="no">Bias</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"red"</span>, <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"dashed"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Bias"</span>)
-<span class="fu"><a href="https://rdrr.io/pkg/cowplot/man/plot_grid.html">plot_grid</a></span>(<span class="kw">plotlist</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="no">p1</span>, <span class="no">p2</span>), <span class="kw">ncol</span> <span class="kw">=</span> <span class="fl">2</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-7-1.png" width="700"></p>
-<p>The approximation is quite good for the standard deviations, but a little too small for the biases.</p>
-</div>
-<div id="the-shakespeare-data" class="section level2">
-<h2 class="hasAnchor">
-<a href="#the-shakespeare-data" class="anchor"></a>The Shakespeare data</h2>
-<p>Here we are given the word counts for the entire Shakespeare canon in the data set <code>bardWordCount</code>. We assume the <span class="math inline">\(i\)</span>th distinct word appeared <span class="math inline">\(X_i \sim Poisson(\Theta_i)\)</span> times in the canon.</p>
-<div class="sourceCode" id="cb7"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/utils/data.html">data</a></span>(<span class="no">bardWordCount</span>)
-<span class="fu"><a href="https://rdrr.io/r/utils/str.html">str</a></span>(<span class="no">bardWordCount</span>)</pre></body></html></div>
-<pre><code>##  num [1:100] 14376 4343 2292 1463 1043 ...</code></pre>
-<p>We take the support set <span class="math inline">\(\mathcal{T}\)</span> for <span class="math inline">\(\Theta\)</span> to be equally spaced on the log-scale and the sample space for <span class="math inline">\(\mathcal{X}\)</span> to be <span class="math inline">\((1,2,\ldots,100).\)</span></p>
-<div class="sourceCode" id="cb9"><html><body><pre class="r"><span class="no">lambda</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(-<span class="fl">4</span>, <span class="fl">4.5</span>, <span class="fl">.025</span>)
-<span class="no">tau</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/Log.html">exp</a></span>(<span class="no">lambda</span>)</pre></body></html></div>
-<p>Using a regularization parameter of <code>c0=2</code> we can deconvolve the data to get <span class="math inline">\(\hat{g}.\)</span></p>
-<div class="sourceCode" id="cb10"><html><body><pre class="r"><span class="no">result</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">bardWordCount</span>, <span class="kw">n</span> <span class="kw">=</span> <span class="fl">100</span>, <span class="kw">c0</span><span class="kw">=</span><span class="fl">2</span>)
-<span class="no">stats</span> <span class="kw">&lt;-</span> <span class="no">result</span>$<span class="no">stats</span></pre></body></html></div>
-<p>The plot below shows the Empirical Bayes deconvoluation estimates for the Shakespeare word counts.</p>
-<div class="sourceCode" id="cb11"><html><body><pre class="r"><span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>() +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">stats</span>[, <span class="st">"g"</span>])) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/Log.html">log</a></span>(<span class="no">theta</span>)), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu">g</span>(<span class="no">theta</span>)))</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-11-1.png" width="700"></p>
-<p>The quantity <span class="math inline">\(R(\alpha)\)</span> in the paper (Efron, Biometrika 2015) can be extracted from the <code>stats</code> list; in this case for a regularization parameter of <code>c0=2</code> we can print its value:</p>
-<div class="sourceCode" id="cb12"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/print.html">print</a></span>(<span class="no">result</span>$<span class="no">S</span>)</pre></body></html></div>
-<pre><code>## [1] 0.005534954</code></pre>
-<p>The <code>stats</code> list contains other estimates quantities as well.</p>
-<p>As noted in the paper citing this package, about 44 percent of the total mass of <span class="math inline">\(\hat{g}\)</span> lies below <span class="math inline">\(\Theta = 1\)</span>, which is an underestimate. This can be corrected for by defining <span class="math display">\[
-\tilde{g} = c_1\hat{g} / (1 - e^{-\theta_j}),
-\]</span> where <span class="math inline">\(c_1\)</span> is the constant that normalizes <span class="math inline">\(\tilde{g}\)</span>.</p>
-<p>When there is truncation at zero, as is the case here, the <code>deconvolveR</code> package now returns an additional column in <code>stats[, "tg"]</code> which contains this correction for <em>thinning</em>. (The default invocation of <code>deconv</code> assumes zero truncation for the Poisson family, argument <code>ignoreZero = FALSE</code>).</p>
-<div class="sourceCode" id="cb14"><html><body><pre class="r"><span class="no">d</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(<span class="kw">lambda</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">g</span> <span class="kw">=</span> <span class="no">stats</span>[, <span class="st">"g"</span>], <span class="kw">tg</span> <span class="kw">=</span> <span class="no">stats</span>[, <span class="st">"tg"</span>], <span class="kw">SE.g</span> <span class="kw">=</span> <span class="no">stats</span>[, <span class="st">"SE.g"</span>])
-<span class="no">indices</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="fl">1</span>, <span class="fu"><a href="https://rdrr.io/r/base/length.html">length</a></span>(<span class="no">lambda</span>), <span class="fl">5</span>)
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="no">d</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_linerange.html">geom_errorbar</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="no">d</span>[<span class="no">indices</span>, ],
-                  <span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">ymin</span> <span class="kw">=</span> <span class="no">g</span> - <span class="no">SE.g</span>, <span class="kw">ymax</span> <span class="kw">=</span> <span class="no">g</span> + <span class="no">SE.g</span>),
-                  <span class="kw">width</span> <span class="kw">=</span> <span class="fl">.01</span>, <span class="kw">color</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/Log.html">log</a></span>(<span class="no">theta</span>)), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu">g</span>(<span class="no">theta</span>))) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/lims.html">ylim</a></span>(<span class="fl">0</span>, <span class="fl">0.006</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">tg</span>), <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"dashed"</span>, <span class="kw">color</span> <span class="kw">=</span> <span class="st">"red"</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-13-1.png" width="700"></p>
-<p>We can now plot the posterior estimates.</p>
-<div class="sourceCode" id="cb15"><html><body><pre class="r"><span class="no">gPost</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq_len</a></span>(<span class="fl">100</span>), <span class="kw">function</span>(<span class="no">i</span>) <span class="fu"><a href="https://rdrr.io/r/base/eval.html">local</a></span>({<span class="no">tg</span> <span class="kw">&lt;-</span> <span class="no">d</span>$<span class="no">tg</span> * <span class="no">result</span>$<span class="no">P</span>[<span class="no">i</span>, ]; <span class="no">tg</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">tg</span>)}))
-<span class="no">plots</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">lapply</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">1</span>, <span class="fl">2</span>, <span class="fl">4</span>, <span class="fl">8</span>), <span class="kw">function</span>(<span class="no">i</span>) {
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>() +
-        <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">gPost</span>[, <span class="no">i</span>])) +
-        <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu">g</span>(<span class="no">theta</span>)),
-             <span class="kw">title</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/sprintf.html">sprintf</a></span>(<span class="st">"x = %d"</span>, <span class="no">i</span>))
-})
-<span class="no">plots</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/funprog.html">Map</a></span>(<span class="kw">f</span> <span class="kw">=</span> <span class="kw">function</span>(<span class="no">p</span>, <span class="no">xlim</span>) <span class="no">p</span> + <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/lims.html">xlim</a></span>(<span class="fl">0</span>, <span class="no">xlim</span>), <span class="no">plots</span>, <span class="fu"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="fl">6</span>, <span class="fl">8</span>, <span class="fl">14</span>, <span class="fl">20</span>))
-<span class="fu"><a href="https://rdrr.io/pkg/cowplot/man/plot_grid.html">plot_grid</a></span>(<span class="kw">plotlist</span> <span class="kw">=</span> <span class="no">plots</span>, <span class="kw">ncol</span> <span class="kw">=</span> <span class="fl">2</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-14-1.png" width="720"></p>
-<div id="bootstrap-comparison" class="section level3">
-<h3 class="hasAnchor">
-<a href="#bootstrap-comparison" class="anchor"></a>Bootstrap Comparison</h3>
-<p>As a check, one can perform a parametric bootstrap using <span class="math display">\[
-y^{*} \sim Mult_n(N, {\mathbf f})
-\]</span> with the MLE <span class="math inline">\(\hat{\mathbf f} = {\mathbf f}(\hat{\alpha})\)</span>. We use 200 bootstrap replcates to compute the standard errors for <span class="math inline">\(\hat{g}\)</span> and compare them to the theoretical values in the plot below. The agreement is pretty good.</p>
-<div class="sourceCode" id="cb16"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/Random.html">set.seed</a></span>(<span class="fl">1783</span>)
-<span class="no">B</span> <span class="kw">&lt;-</span> <span class="fl">200</span>
-<span class="no">fHat</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/numeric.html">as.numeric</a></span>(<span class="no">result</span>$<span class="no">P</span> <span class="kw">%*%</span> <span class="no">d</span>$<span class="no">g</span>)
-<span class="no">fHat</span> <span class="kw">&lt;-</span> <span class="no">fHat</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">fHat</span>)
-<span class="no">yStar</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Multinom.html">rmultinom</a></span>(<span class="kw">n</span> <span class="kw">=</span> <span class="no">B</span>, <span class="kw">size</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">bardWordCount</span>), <span class="kw">prob</span> <span class="kw">=</span> <span class="no">fHat</span>)
-<span class="no">gBoot</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">yStar</span>, <span class="fl">2</span>,
-               <span class="kw">function</span>(<span class="no">y</span>) <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">y</span>, <span class="kw">n</span> <span class="kw">=</span> <span class="fl">100</span>, <span class="kw">c0</span> <span class="kw">=</span> <span class="fl">2</span>)$<span class="no">stats</span>[, <span class="st">"g"</span>])
-<span class="no">seG</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">gBoot</span>, <span class="fl">1</span>, <span class="no">sd</span>)
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="no">d</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">SE.g</span>,
-                            <span class="kw">color</span> <span class="kw">=</span> <span class="st">"Theoretical"</span>, <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"Theoretical"</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">lambda</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">seG</span>,
-                            <span class="kw">color</span> <span class="kw">=</span> <span class="st">"Bootstrap"</span>, <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"Bootstrap"</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/scale_manual.html">scale_color_manual</a></span>(<span class="kw">name</span> <span class="kw">=</span> <span class="st">"Legend"</span>,
-                       <span class="kw">values</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"Bootstrap"</span> <span class="kw">=</span> <span class="st">"black"</span>, <span class="st">"Theoretical"</span> <span class="kw">=</span> <span class="st">"red"</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/scale_manual.html">scale_linetype_manual</a></span>(<span class="kw">name</span> <span class="kw">=</span> <span class="st">"Legend"</span>,
-                          <span class="kw">values</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"Bootstrap"</span> <span class="kw">=</span> <span class="st">"solid"</span>, <span class="st">"Theoretical"</span> <span class="kw">=</span> <span class="st">"dashed"</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/theme.html">theme</a></span>(<span class="kw">legend.title</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/element.html">element_blank</a></span>()) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/Log.html">log</a></span>(<span class="no">theta</span>)), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu"><a href="https://rdrr.io/r/stats/sigma.html">sigma</a></span>(<span class="fu"><a href="https://rdrr.io/r/stats/influence.measures.html">hat</a></span>(<span class="no">g</span>))))</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-15-1.png" width="700"></p>
-</div>
-<div id="predict-ratio-of-new-distinct-words" class="section level3">
-<h3 class="hasAnchor">
-<a href="#predict-ratio-of-new-distinct-words" class="anchor"></a>Predict ratio of new distinct words</h3>
-<p>Suppose then that a previously unknown Shakespearean corpus of length <span class="math inline">\(t\times C\)</span> were found, <span class="math inline">\(C\times 900,000\)</span> the length of the known canon. Assuming a Poisson process model with intensity <span class="math inline">\(\Theta_i\)</span> for word <span class="math inline">\(i\)</span>, the probability that word <span class="math inline">\(i\)</span> did not appear in the canon but does appear in the new corpus is <span class="math display">\[
-e^{-\Theta_i}\left(1-e^{-\Theta_it}\right);
-\]</span> yielding after some work, an estimate for <span class="math inline">\(R(t)\)</span>, the expected number of distinct new words found, divided by <span class="math inline">\(N\)</span>, the observed number of distinct words in the canon: <span class="math display">\[
-R(t)=\sum_{j=1}^m\hat{g}_jr_j(t),
-\]</span> with <span class="math display">\[
-r_j=\frac{e^{-\theta_{(j)}}}{1-e^{-\theta_{(j)}}}\left(1-e^{-\theta_{(j)}t}\right).
-\]</span></p>
-<p>We can compute the <span class="math inline">\(R(t)\)</span> and plot it as follows.</p>
-<div class="sourceCode" id="cb17"><html><body><pre class="r"><span class="no">gHat</span> <span class="kw">&lt;-</span> <span class="no">stats</span>[, <span class="st">"g"</span>]
-<span class="no">Rfn</span> <span class="kw">&lt;-</span> <span class="kw">function</span>(<span class="no">t</span>) {
-    <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>( <span class="no">gHat</span> * (<span class="fl">1</span> - <span class="fu"><a href="https://rdrr.io/r/base/Log.html">exp</a></span>(-<span class="no">tau</span> * <span class="no">t</span>)) / (<span class="fu"><a href="https://rdrr.io/r/base/Log.html">exp</a></span>(<span class="no">tau</span>) - <span class="fl">1</span>) )
-}
-<span class="no">r</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="fl">0</span>:<span class="fl">10</span>, <span class="no">Rfn</span>)
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>() +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fl">0</span>:<span class="fl">10</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">r</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="st">"time multiple t"</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu">R</span>(<span class="no">t</span>)))</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-16-1.png" width="700"></p>
-<p>And the (speculative) doubling time for Shakespeare’s vocabulary is easy to compute too.</p>
-<div class="sourceCode" id="cb18"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/print.html">print</a></span>(<span class="fu"><a href="https://rdrr.io/r/stats/uniroot.html">uniroot</a></span>(<span class="kw">f</span> <span class="kw">=</span> <span class="kw">function</span>(<span class="no">x</span>) <span class="fu">Rfn</span>(<span class="no">x</span>) - <span class="fl">1</span>, <span class="kw">interval</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">2</span>, <span class="fl">4</span>))$<span class="no">root</span>)</pre></body></html></div>
-<pre><code>## [1] 3.653132</code></pre>
-</div>
-</div>
-<div id="normal-example" class="section level2">
-<h2 class="hasAnchor">
-<a href="#normal-example" class="anchor"></a>Normal Example</h2>
-<p>Consider a data set that is generated via the following mechanism.</p>
-<p><span class="math display">\[
-z_i \sim N(\mu_i, 1), \mbox{ $i = 1,2,\ldots, N = 10,000$}
-\]</span></p>
-<p>with</p>
-<p><span class="math display">\[
-\mu_i=
-    \begin{cases}
-        0,        &amp; \mbox{with probability .9}\\
-        N(-3, 1), &amp; \mbox{with probability .1}\\
-    \end{cases}
-\]</span></p>
-<div class="sourceCode" id="cb20"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/Random.html">set.seed</a></span>(<span class="fl">129023</span>)
-<span class="no">N</span> <span class="kw">&lt;-</span> <span class="fl">10000</span>
-<span class="no">pi0</span> <span class="kw">&lt;-</span> <span class="fl">.90</span>
-<span class="no">data</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/eval.html">local</a></span>({
-    <span class="no">nullCase</span> <span class="kw">&lt;-</span> (<span class="fu"><a href="https://rdrr.io/r/stats/Uniform.html">runif</a></span>(<span class="no">N</span>) <span class="kw">&lt;=</span> <span class="no">pi0</span>)
-    <span class="no">muAndZ</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/t.html">t</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">nullCase</span>, <span class="kw">function</span>(<span class="no">isNull</span>) {
-        <span class="kw">if</span> (<span class="no">isNull</span>) {
-            <span class="no">mu</span> <span class="kw">&lt;-</span> <span class="fl">0</span>
-            <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="no">mu</span>, <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">rnorm</a></span>(<span class="fl">1</span>))
-        } <span class="kw">else</span> {
-            <span class="no">mu</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">rnorm</a></span>(<span class="fl">1</span>, <span class="kw">mean</span> <span class="kw">=</span> -<span class="fl">3</span>)
-            <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="no">mu</span>, <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">rnorm</a></span>(<span class="fl">1</span>, <span class="kw">mean</span> <span class="kw">=</span> <span class="no">mu</span>))
-        }
-    }))
-    <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(<span class="kw">nullCase</span> <span class="kw">=</span> <span class="no">nullCase</span>, <span class="kw">mu</span> <span class="kw">=</span> <span class="no">muAndZ</span>[, <span class="fl">1</span>], <span class="kw">z</span> <span class="kw">=</span> <span class="no">muAndZ</span>[, <span class="fl">2</span>])
-})</pre></body></html></div>
-<p>Below is a histogram of the data (<span class="math inline">\(z\)</span> values) and that of the <span class="math inline">\(\Theta\)</span>s.</p>
-<div class="sourceCode" id="cb21"><html><body><pre class="r"><span class="no">p1</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">data</span>$<span class="no">z</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_histogram.html">geom_histogram</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">y</span>  <span class="kw">=</span> <span class="no">..count..</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">..count..</span>) ),
-                   <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>, <span class="kw">bins</span> <span class="kw">=</span> <span class="fl">60</span>, <span class="kw">alpha</span> <span class="kw">=</span> <span class="fl">0.5</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="st">"z"</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Density"</span>)
-<span class="no">p2</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">data</span>$<span class="no">mu</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_histogram.html">geom_histogram</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">y</span>  <span class="kw">=</span> <span class="no">..count..</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">..count..</span>) ),
-                   <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>, <span class="kw">bins</span> <span class="kw">=</span> <span class="fl">60</span>, <span class="kw">alpha</span> <span class="kw">=</span> <span class="fl">0.5</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Density"</span>)
-<span class="fu"><a href="https://rdrr.io/pkg/cowplot/man/plot_grid.html">plot_grid</a></span>(<span class="kw">plotlist</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="no">p1</span>, <span class="no">p2</span>), <span class="kw">ncol</span> <span class="kw">=</span> <span class="fl">2</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-19-1.png" width="700"></p>
-<p>Now we deconvolve this using <span class="math inline">\(\mathcal{T} = (-6, -5.75,\ldots, 3)\)</span> and a spike at zero and a fifth-degree polynomial.</p>
-<div class="sourceCode" id="cb22"><html><body><pre class="r"><span class="no">tau</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="kw">from</span> <span class="kw">=</span> -<span class="fl">6</span>, <span class="kw">to</span> <span class="kw">=</span> <span class="fl">3</span>, <span class="kw">by</span> <span class="kw">=</span> <span class="fl">0.25</span>)
-<span class="no">atomIndex</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/which.html">which</a></span>(<span class="no">tau</span> <span class="kw">==</span> <span class="fl">0</span>)
-<span class="no">result</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">data</span>$<span class="no">z</span>, <span class="kw">deltaAt</span> <span class="kw">=</span> <span class="fl">0</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Normal"</span>, <span class="kw">pDegree</span> <span class="kw">=</span> <span class="fl">5</span>)</pre></body></html></div>
-<p>The estimates and the standard errors of the penalized MLE <span class="math inline">\(\hat{g}\)</span> with <code>c0 = 1</code> are shown below.</p>
-<div class="sourceCode" id="cb23"><html><body><pre class="r"><span class="kw pkg">knitr</span><span class="kw ns">::</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html">kable</a></span>(<span class="no">result</span>$<span class="no">stats</span>)</pre></body></html></div>
-<table class="table">
-<thead><tr class="header">
-<th align="right">theta</th>
-<th align="right">g</th>
-<th align="right">SE.g</th>
-<th align="right">G</th>
-<th align="right">SE.G</th>
-<th align="right">Bias.g</th>
-</tr></thead>
-<tbody>
-<tr class="odd">
-<td align="right">-6.00</td>
-<td align="right">0.0001755</td>
-<td align="right">0.0001139</td>
-<td align="right">0.0001755</td>
-<td align="right">0.0001139</td>
-<td align="right">0.0002100</td>
-</tr>
-<tr class="even">
-<td align="right">-5.75</td>
-<td align="right">0.0003137</td>
-<td align="right">0.0001627</td>
-<td align="right">0.0004892</td>
-<td align="right">0.0002764</td>
-<td align="right">0.0002944</td>
-</tr>
-<tr class="odd">
-<td align="right">-5.50</td>
-<td align="right">0.0005542</td>
-<td align="right">0.0002188</td>
-<td align="right">0.0010434</td>
-<td align="right">0.0004943</td>
-<td align="right">0.0003817</td>
-</tr>
-<tr class="even">
-<td align="right">-5.25</td>
-<td align="right">0.0009558</td>
-<td align="right">0.0002710</td>
-<td align="right">0.0019991</td>
-<td align="right">0.0007611</td>
-<td align="right">0.0004355</td>
-</tr>
-<tr class="odd">
-<td align="right">-5.00</td>
-<td align="right">0.0015902</td>
-<td align="right">0.0003068</td>
-<td align="right">0.0035893</td>
-<td align="right">0.0010496</td>
-<td align="right">0.0003936</td>
-</tr>
-<tr class="even">
-<td align="right">-4.75</td>
-<td align="right">0.0025218</td>
-<td align="right">0.0003396</td>
-<td align="right">0.0061111</td>
-<td align="right">0.0013181</td>
-<td align="right">0.0001835</td>
-</tr>
-<tr class="odd">
-<td align="right">-4.50</td>
-<td align="right">0.0037665</td>
-<td align="right">0.0004384</td>
-<td align="right">0.0098775</td>
-<td align="right">0.0015480</td>
-<td align="right">-0.0002271</td>
-</tr>
-<tr class="even">
-<td align="right">-4.25</td>
-<td align="right">0.0052352</td>
-<td align="right">0.0006237</td>
-<td align="right">0.0151127</td>
-<td align="right">0.0017954</td>
-<td align="right">-0.0007502</td>
-</tr>
-<tr class="odd">
-<td align="right">-4.00</td>
-<td align="right">0.0067030</td>
-<td align="right">0.0007782</td>
-<td align="right">0.0218157</td>
-<td align="right">0.0021384</td>
-<td align="right">-0.0011392</td>
-</tr>
-<tr class="even">
-<td align="right">-3.75</td>
-<td align="right">0.0079323</td>
-<td align="right">0.0008168</td>
-<td align="right">0.0297481</td>
-<td align="right">0.0025338</td>
-<td align="right">-0.0011786</td>
-</tr>
-<tr class="odd">
-<td align="right">-3.50</td>
-<td align="right">0.0087575</td>
-<td align="right">0.0007875</td>
-<td align="right">0.0385055</td>
-<td align="right">0.0028572</td>
-<td align="right">-0.0008249</td>
-</tr>
-<tr class="even">
-<td align="right">-3.25</td>
-<td align="right">0.0091050</td>
-<td align="right">0.0008190</td>
-<td align="right">0.0476105</td>
-<td align="right">0.0030494</td>
-<td align="right">-0.0001984</td>
-</tr>
-<tr class="odd">
-<td align="right">-3.00</td>
-<td align="right">0.0089988</td>
-<td align="right">0.0009356</td>
-<td align="right">0.0566093</td>
-<td align="right">0.0031703</td>
-<td align="right">0.0004827</td>
-</tr>
-<tr class="even">
-<td align="right">-2.75</td>
-<td align="right">0.0085341</td>
-<td align="right">0.0010240</td>
-<td align="right">0.0651434</td>
-<td align="right">0.0033244</td>
-<td align="right">0.0009991</td>
-</tr>
-<tr class="odd">
-<td align="right">-2.50</td>
-<td align="right">0.0078394</td>
-<td align="right">0.0010049</td>
-<td align="right">0.0729829</td>
-<td align="right">0.0035304</td>
-<td align="right">0.0012035</td>
-</tr>
-<tr class="even">
-<td align="right">-2.25</td>
-<td align="right">0.0070391</td>
-<td align="right">0.0009187</td>
-<td align="right">0.0800220</td>
-<td align="right">0.0036975</td>
-<td align="right">0.0010485</td>
-</tr>
-<tr class="odd">
-<td align="right">-2.00</td>
-<td align="right">0.0062096</td>
-<td align="right">0.0009410</td>
-<td align="right">0.0862316</td>
-<td align="right">0.0037737</td>
-<td align="right">0.0006476</td>
-</tr>
-<tr class="even">
-<td align="right">-1.75</td>
-<td align="right">0.0053923</td>
-<td align="right">0.0011197</td>
-<td align="right">0.0916239</td>
-<td align="right">0.0038631</td>
-<td align="right">0.0001722</td>
-</tr>
-<tr class="odd">
-<td align="right">-1.50</td>
-<td align="right">0.0046182</td>
-<td align="right">0.0013216</td>
-<td align="right">0.0962421</td>
-<td align="right">0.0041634</td>
-<td align="right">-0.0002559</td>
-</tr>
-<tr class="even">
-<td align="right">-1.25</td>
-<td align="right">0.0039083</td>
-<td align="right">0.0014443</td>
-<td align="right">0.1001503</td>
-<td align="right">0.0047866</td>
-<td align="right">-0.0005669</td>
-</tr>
-<tr class="odd">
-<td align="right">-1.00</td>
-<td align="right">0.0032743</td>
-<td align="right">0.0014544</td>
-<td align="right">0.1034247</td>
-<td align="right">0.0056648</td>
-<td align="right">-0.0007344</td>
-</tr>
-<tr class="even">
-<td align="right">-0.75</td>
-<td align="right">0.0027209</td>
-<td align="right">0.0013594</td>
-<td align="right">0.1061456</td>
-<td align="right">0.0066420</td>
-<td align="right">-0.0007631</td>
-</tr>
-<tr class="odd">
-<td align="right">-0.50</td>
-<td align="right">0.0022468</td>
-<td align="right">0.0011884</td>
-<td align="right">0.1083923</td>
-<td align="right">0.0075782</td>
-<td align="right">-0.0006763</td>
-</tr>
-<tr class="even">
-<td align="right">-0.25</td>
-<td align="right">0.0018442</td>
-<td align="right">0.0009889</td>
-<td align="right">0.1102365</td>
-<td align="right">0.0083898</td>
-<td align="right">-0.0005154</td>
-</tr>
-<tr class="odd">
-<td align="right">0.00</td>
-<td align="right">0.8846685</td>
-<td align="right">0.0103603</td>
-<td align="right">0.9949050</td>
-<td align="right">0.0032481</td>
-<td align="right">-0.0007798</td>
-</tr>
-<tr class="even">
-<td align="right">0.25</td>
-<td align="right">0.0012121</td>
-<td align="right">0.0006634</td>
-<td align="right">0.9961171</td>
-<td align="right">0.0026657</td>
-<td align="right">-0.0001414</td>
-</tr>
-<tr class="odd">
-<td align="right">0.50</td>
-<td align="right">0.0009670</td>
-<td align="right">0.0005580</td>
-<td align="right">0.9970841</td>
-<td align="right">0.0021484</td>
-<td align="right">0.0000199</td>
-</tr>
-<tr class="even">
-<td align="right">0.75</td>
-<td align="right">0.0007616</td>
-<td align="right">0.0004773</td>
-<td align="right">0.9978457</td>
-<td align="right">0.0016891</td>
-<td align="right">0.0001464</td>
-</tr>
-<tr class="odd">
-<td align="right">1.00</td>
-<td align="right">0.0005909</td>
-<td align="right">0.0004069</td>
-<td align="right">0.9984366</td>
-<td align="right">0.0012899</td>
-<td align="right">0.0002343</td>
-</tr>
-<tr class="even">
-<td align="right">1.25</td>
-<td align="right">0.0004510</td>
-<td align="right">0.0003385</td>
-<td align="right">0.9988876</td>
-<td align="right">0.0009551</td>
-<td align="right">0.0002847</td>
-</tr>
-<tr class="odd">
-<td align="right">1.50</td>
-<td align="right">0.0003383</td>
-<td align="right">0.0002718</td>
-<td align="right">0.9992259</td>
-<td align="right">0.0006855</td>
-<td align="right">0.0003024</td>
-</tr>
-<tr class="even">
-<td align="right">1.75</td>
-<td align="right">0.0002500</td>
-<td align="right">0.0002114</td>
-<td align="right">0.9994759</td>
-<td align="right">0.0004756</td>
-<td align="right">0.0002951</td>
-</tr>
-<tr class="odd">
-<td align="right">2.00</td>
-<td align="right">0.0001824</td>
-<td align="right">0.0001602</td>
-<td align="right">0.9996583</td>
-<td align="right">0.0003163</td>
-<td align="right">0.0002713</td>
-</tr>
-<tr class="even">
-<td align="right">2.25</td>
-<td align="right">0.0001318</td>
-<td align="right">0.0001191</td>
-<td align="right">0.9997901</td>
-<td align="right">0.0001977</td>
-<td align="right">0.0002386</td>
-</tr>
-<tr class="odd">
-<td align="right">2.50</td>
-<td align="right">0.0000945</td>
-<td align="right">0.0000875</td>
-<td align="right">0.9998846</td>
-<td align="right">0.0001104</td>
-<td align="right">0.0002027</td>
-</tr>
-<tr class="even">
-<td align="right">2.75</td>
-<td align="right">0.0000674</td>
-<td align="right">0.0000639</td>
-<td align="right">0.9999520</td>
-<td align="right">0.0000466</td>
-<td align="right">0.0001677</td>
-</tr>
-<tr class="odd">
-<td align="right">3.00</td>
-<td align="right">0.0000480</td>
-<td align="right">0.0000466</td>
-<td align="right">1.0000000</td>
-<td align="right">0.0000000</td>
-<td align="right">0.0001360</td>
-</tr>
-</tbody>
-</table>
-<p>Per the above table, the estimated probability of <span class="math inline">\(\mu = 0\)</span> is 0.885 <span class="math inline">\(\pm\)</span> 0.01 with a bias of about -0.001.</p>
-<p>We can now plot the <span class="math inline">\(g\)</span>-estimate removing the atom at 0.</p>
-<div class="sourceCode" id="cb24"><html><body><pre class="r"><span class="no">gData</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/as.data.frame.html">as.data.frame</a></span>(<span class="no">result</span>$<span class="no">stats</span>[-<span class="no">atomIndex</span>, <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"theta"</span>, <span class="st">"g"</span>)])
-<span class="no">gData</span>$<span class="no">g</span> <span class="kw">&lt;-</span> <span class="no">gData</span>$<span class="no">g</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">gData</span>$<span class="no">g</span>)
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="no">gData</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">dnorm</a></span>(<span class="no">theta</span>, <span class="kw">mean</span> <span class="kw">=</span> -<span class="fl">3</span>)),
-                            <span class="kw">color</span> <span class="kw">=</span> <span class="st">"red"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu">g</span>(<span class="no">theta</span>)))</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-22-1.png" width="700"></p>
-<p>The density approximation is not accurate at all, however, the posterior estimates for the g’s are similar to what one obtains by the Benjamini-Yekutieli procedure as shown below.</p>
-<ol style="list-style-type: decimal">
-<li>Sort the <span class="math inline">\(p\)</span>-values.</li>
-</ol>
-<div class="sourceCode" id="cb25"><html><body><pre class="r"><span class="no">p</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">pnorm</a></span>(<span class="no">data</span>$<span class="no">z</span>)
-<span class="no">orderP</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/order.html">order</a></span>(<span class="no">p</span>)
-<span class="no">p</span> <span class="kw">&lt;-</span> <span class="no">p</span>[<span class="no">orderP</span>]</pre></body></html></div>
-<ol start="2" style="list-style-type: decimal">
-<li>Compute <span class="math inline">\(R\)</span>, the number of discoveries and count the number of false discoveries</li>
-</ol>
-<div class="sourceCode" id="cb26"><html><body><pre class="r"><span class="co">## FCR</span>
-<span class="no">q</span> <span class="kw">&lt;-</span> <span class="fl">0.05</span>
-<span class="no">R</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/Extremes.html">max</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/which.html">which</a></span>(<span class="no">p</span> <span class="kw">&lt;=</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq_len</a></span>(<span class="no">N</span>) * <span class="no">q</span> / <span class="no">N</span>))
-<span class="no">discIdx</span> <span class="kw">&lt;-</span> <span class="no">orderP</span>[<span class="fl">1</span>:<span class="no">R</span>]
-<span class="no">disc</span> <span class="kw">&lt;-</span> <span class="no">data</span>[<span class="no">discIdx</span>, ]
-<span class="fu"><a href="https://rdrr.io/r/base/cat.html">cat</a></span>(<span class="st">"BY_q procedure discoveries"</span>, <span class="no">R</span>, <span class="st">"cases,"</span>, <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">disc</span>$<span class="no">nullCase</span>),
-    <span class="st">"actual nulls among them.\n"</span>)</pre></body></html></div>
-<pre><code>## BY_q procedure discoveries 610 cases, 22 actual nulls among them.</code></pre>
-<ol start="3" style="list-style-type: decimal">
-<li>Construct Benjamini-Yekutieli and Bayes confidence intervals.</li>
-</ol>
-<div class="sourceCode" id="cb28"><html><body><pre class="r"><span class="no">alphaR</span> <span class="kw">&lt;-</span> <span class="fl">1</span> - <span class="no">R</span> * <span class="no">q</span> / <span class="no">N</span>
-<span class="no">zAlpha</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">qnorm</a></span>(<span class="no">alphaR</span>, <span class="kw">lower.tail</span> <span class="kw">=</span> <span class="fl">FALSE</span>)
-<span class="no">zMarker</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/Extremes.html">max</a></span>(<span class="no">disc</span>$<span class="no">z</span>)
-<span class="no">xlim</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(-<span class="fl">7.6</span>, <span class="fl">0.0</span>)
-<span class="no">ylim</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(-<span class="fl">10</span>, <span class="fl">0.0</span>)
-<span class="no">BY.lo</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="no">xlim</span>[<span class="fl">1</span>] - <span class="no">zAlpha</span>, <span class="no">xlim</span>[<span class="fl">2</span>] - <span class="no">zAlpha</span>)
-<span class="no">BY.up</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="no">xlim</span>[<span class="fl">1</span>] + <span class="no">zAlpha</span>, <span class="no">xlim</span>[<span class="fl">2</span>] + <span class="no">zAlpha</span>)
-<span class="no">Bayes.lo</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">0.5</span> * (<span class="no">xlim</span>[<span class="fl">1</span>] - <span class="fl">3</span>) - <span class="fl">1.96</span> / <span class="fu"><a href="https://rdrr.io/r/base/MathFun.html">sqrt</a></span>(<span class="fl">2</span>), <span class="fl">0.5</span> * (<span class="no">xlim</span>[<span class="fl">2</span>] - <span class="fl">3</span>) - <span class="fl">1.96</span> / <span class="fu"><a href="https://rdrr.io/r/base/MathFun.html">sqrt</a></span>(<span class="fl">2</span>))
-<span class="no">Bayes.up</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">0.5</span> * (<span class="no">xlim</span>[<span class="fl">1</span>] - <span class="fl">3</span>) + <span class="fl">1.96</span> / <span class="fu"><a href="https://rdrr.io/r/base/MathFun.html">sqrt</a></span>(<span class="fl">2</span>), <span class="fl">0.5</span> * (<span class="no">xlim</span>[<span class="fl">2</span>] - <span class="fl">3</span>) + <span class="fl">1.96</span> / <span class="fu"><a href="https://rdrr.io/r/base/MathFun.html">sqrt</a></span>(<span class="fl">2</span>))</pre></body></html></div>
-<ol start="4" style="list-style-type: decimal">
-<li>Compute the estimated posterior density for of <span class="math inline">\(\mu\)</span> given <span class="math inline">\(z\)</span> and construct the 95% credible intervals for <span class="math inline">\(\mu\)</span> given <span class="math inline">\(z\)</span>.</li>
-</ol>
-<div class="sourceCode" id="cb29"><html><body><pre class="r"><span class="no">d</span> <span class="kw">&lt;-</span> <span class="no">data</span>[<span class="fu"><a href="https://rdrr.io/r/base/order.html">order</a></span>(<span class="no">data</span>$<span class="no">mu</span>), ]
-<span class="no">muVals</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/unique.html">unique</a></span>(<span class="no">d</span>$<span class="no">mu</span>)
-<span class="no">s</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/as.data.frame.html">as.data.frame</a></span>(<span class="no">result</span>$<span class="no">stats</span>)
-<span class="no">indices</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/findInterval.html">findInterval</a></span>(<span class="no">muVals</span>, <span class="no">s</span>$<span class="no">theta</span>) + <span class="fl">1</span>
-<span class="no">gMu</span> <span class="kw">&lt;-</span> <span class="no">s</span>$<span class="no">g</span>[<span class="no">indices</span>]
-<span class="no">st</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/Extremes.html">min</a></span>(<span class="no">data</span>$<span class="no">z</span>), -<span class="fl">2.0</span>, <span class="kw">length.out</span> <span class="kw">=</span> <span class="fl">40</span>)
-<span class="no">gMuPhi</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">st</span>, <span class="kw">function</span>(<span class="no">z</span>) <span class="no">gMu</span> * <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">dnorm</a></span>(<span class="no">z</span> - <span class="no">muVals</span>))
-<span class="no">g2</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">gMuPhi</span>, <span class="fl">2</span>, <span class="kw">function</span>(<span class="no">x</span>) <span class="fu"><a href="https://rdrr.io/r/base/cumsum.html">cumsum</a></span>(<span class="no">x</span>)/<span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">x</span>))
-<span class="no">pct</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">g2</span>, <span class="fl">2</span>, <span class="kw">function</span>(<span class="no">dist</span>) <span class="fu"><a href="https://rdrr.io/r/stats/approxfun.html">approx</a></span>(<span class="kw">y</span> <span class="kw">=</span> <span class="no">muVals</span>, <span class="kw">x</span> <span class="kw">=</span> <span class="no">dist</span>, <span class="kw">xout</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">0.025</span>, <span class="fl">0.975</span>)))
-<span class="no">qVals</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">pct</span>, <span class="kw">function</span>(<span class="no">item</span>) <span class="no">item</span>$<span class="no">y</span>)</pre></body></html></div>
-<ol start="5" style="list-style-type: decimal">
-<li>Plot it</li>
-</ol>
-<div class="sourceCode" id="cb30"><html><body><pre class="r"><span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>() +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">xlim</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">BY.lo</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">xlim</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">BY.up</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">xlim</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">Bayes.lo</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"magenta"</span>,
-              <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"dashed"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">xlim</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">Bayes.up</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"magenta"</span>,
-              <span class="kw">linetype</span> <span class="kw">=</span> <span class="st">"dashed"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_point.html">geom_point</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">disc</span>$<span class="no">z</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">disc</span>$<span class="no">mu</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"red"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_point.html">geom_point</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">disc</span>$<span class="no">z</span>[<span class="no">disc</span>$<span class="no">nullCase</span>], <span class="kw">y</span> <span class="kw">=</span> <span class="no">disc</span>$<span class="no">mu</span>[<span class="no">disc</span>$<span class="no">nullCase</span>]),
-                             <span class="kw">color</span> <span class="kw">=</span> <span class="st">"orange"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/rep.html">rep</a></span>(<span class="no">zMarker</span>, <span class="fl">2</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(-<span class="fl">10</span>, <span class="fl">1</span>))) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">st</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">qVals</span>[<span class="fl">1</span>, ]), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">st</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">qVals</span>[<span class="fl">2</span>, ]), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="st">"Observed z"</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">mu</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> -<span class="fl">1</span>, <span class="kw">y</span> <span class="kw">=</span> -<span class="fl">4.25</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"BY.lo"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> -<span class="fl">1</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fl">1.25</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"BY.up"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> -<span class="fl">7.5</span>, <span class="kw">y</span> <span class="kw">=</span> -<span class="fl">6.1</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"Bayes.lo"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> -<span class="fl">7.5</span>, <span class="kw">y</span> <span class="kw">=</span> -<span class="fl">3.4</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"Bayes.up"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> -<span class="fl">2.0</span>, <span class="kw">y</span> <span class="kw">=</span> -<span class="fl">1.75</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"EB.lo"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> -<span class="fl">2.0</span>, <span class="kw">y</span> <span class="kw">=</span> -<span class="fl">3.9</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"EB.up"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> <span class="no">zMarker</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fl">1.25</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/character.html">as.character</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/Round.html">round</a></span>(<span class="no">zMarker</span>, <span class="fl">2</span>)))</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-27-1.png" width="700"></p>
-</div>
-<div id="another-normal-example-twin-towers" class="section level2">
-<h2 class="hasAnchor">
-<a href="#another-normal-example-twin-towers" class="anchor"></a>Another Normal example (Twin Towers)</h2>
-<p>In the first normal example, the <span class="math inline">\(\theta\)</span> distribution had a significant atom at 0 and the rest of the density was smeared around -3. We now investigate what happens when the <span class="math inline">\(\theta\)</span> distribution is clearly bimodal. Below is a histogram of the the <span class="math inline">\(\theta\)</span> and alongside a histogram of the data, generated using <span class="math inline">\(X_i \sim N(\theta_i, 1)\)</span>.</p>
-<div class="sourceCode" id="cb31"><html><body><pre class="r"><span class="no">p1</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">disjointTheta</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_histogram.html">geom_histogram</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">y</span>  <span class="kw">=</span> <span class="no">..count..</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">..count..</span>) ),
-                   <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>, <span class="kw">bins</span> <span class="kw">=</span> <span class="fl">60</span>, <span class="kw">alpha</span> <span class="kw">=</span> <span class="fl">0.5</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Density"</span>)
-<span class="fu"><a href="https://rdrr.io/r/base/Random.html">set.seed</a></span> (<span class="fl">2332</span>)
-<span class="no">z</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/stats/Normal.html">rnorm</a></span>(<span class="kw">n</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/length.html">length</a></span>(<span class="no">disjointTheta</span>), <span class="kw">mean</span> <span class="kw">=</span> <span class="no">disjointTheta</span>)
-<span class="no">p2</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">z</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_histogram.html">geom_histogram</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">y</span>  <span class="kw">=</span> <span class="no">..count..</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">..count..</span>) ),
-                   <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>, <span class="kw">bins</span> <span class="kw">=</span> <span class="fl">60</span>, <span class="kw">alpha</span> <span class="kw">=</span> <span class="fl">0.5</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="st">"z"</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Density"</span>)
-<span class="fu"><a href="https://rdrr.io/pkg/cowplot/man/plot_grid.html">plot_grid</a></span>(<span class="kw">plotlist</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="no">p1</span>, <span class="no">p2</span>), <span class="kw">ncol</span> <span class="kw">=</span> <span class="fl">2</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-28-1.png" width="700"></p>
-<p>We deconvolve the data, using various values for the spline degrees of freedom.</p>
-<div class="sourceCode" id="cb32"><html><body><pre class="r"><span class="no">tau</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="kw">from</span> <span class="kw">=</span> -<span class="fl">4</span>, <span class="kw">to</span> <span class="kw">=</span> <span class="fl">6</span>, <span class="kw">by</span> <span class="kw">=</span> <span class="fl">0.2</span>)
-<span class="no">plots1</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">lapply</a></span>(<span class="fl">2</span>:<span class="fl">8</span>,
-                 <span class="kw">function</span>(<span class="no">p</span>) {
-                     <span class="no">result</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">z</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Normal"</span>, <span class="kw">pDegree</span> <span class="kw">=</span> <span class="no">p</span>)
-                     <span class="no">g</span> <span class="kw">&lt;-</span> <span class="no">result</span>$<span class="no">stats</span>[, <span class="st">"g"</span>]
-                     <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">disjointTheta</span>)) +
-                         <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_histogram.html">geom_histogram</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">y</span> <span class="kw">=</span> <span class="no">..count..</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">..count..</span>)),
-                                        <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>, <span class="kw">bins</span> <span class="kw">=</span> <span class="fl">60</span>, <span class="kw">alpha</span> <span class="kw">=</span> <span class="fl">0.2</span>) +
-                         <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-                         <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Density"</span>,
-                              <span class="kw">title</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/sprintf.html">sprintf</a></span>(<span class="st">"DF = %d"</span>, <span class="no">p</span>))
-                })</pre></body></html></div>
-<p>Choosing the degrees of freedom to be 7, we now examine the effect of regularization parameter <span class="math inline">\(c0\)</span>.</p>
-<div class="sourceCode" id="cb33"><html><body><pre class="r"><span class="no">plots2</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">lapply</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">0.5</span>, <span class="fl">1</span>, <span class="fl">2</span>, <span class="fl">4</span>, <span class="fl">8</span>, <span class="fl">16</span>, <span class="fl">32</span>),
-                 <span class="kw">function</span>(<span class="no">c0</span>) {
-                     <span class="no">result</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">z</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Normal"</span>, <span class="kw">pDegree</span> <span class="kw">=</span> <span class="fl">6</span>,
-                                      <span class="kw">c0</span> <span class="kw">=</span> <span class="no">c0</span>)
-                     <span class="no">g</span> <span class="kw">&lt;-</span> <span class="no">result</span>$<span class="no">stats</span>[, <span class="st">"g"</span>]
-                     <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">disjointTheta</span>)) +
-                         <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_histogram.html">geom_histogram</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">y</span> <span class="kw">=</span> <span class="no">..count..</span> / <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">..count..</span>)),
-                                        <span class="kw">color</span> <span class="kw">=</span> <span class="st">"brown"</span>, <span class="kw">bins</span> <span class="kw">=</span> <span class="fl">60</span>, <span class="kw">alpha</span> <span class="kw">=</span> <span class="fl">0.2</span>) +
-                         <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g</span>), <span class="kw">color</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-                         <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="st">"Density"</span>,
-                              <span class="kw">title</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/sprintf.html">sprintf</a></span>(<span class="st">"C0 = %.1f"</span>, <span class="no">c0</span>))
-                })
-<span class="no">plots</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/mapply.html">mapply</a></span>(<span class="kw">function</span>(<span class="no">x</span>, <span class="no">y</span>) <span class="fu"><a href="https://rdrr.io/r/base/list.html">list</a></span>(<span class="no">x</span>, <span class="no">y</span>), <span class="no">plots1</span>, <span class="no">plots2</span>)
-<span class="fu"><a href="https://rdrr.io/pkg/cowplot/man/plot_grid.html">plot_grid</a></span>(<span class="kw">plotlist</span> <span class="kw">=</span> <span class="no">plots</span>, <span class="kw">ncol</span> <span class="kw">=</span> <span class="fl">2</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-30-1.png" width="720"></p>
-<p>The quantity <span class="math inline">\(S(\alpha)\)</span> is a measure of the strength of the penalty term <span class="math inline">\(c_0\)</span> and is returned by <code>deconv</code> in the result list as a named item <code>S</code>. Below we vary <span class="math inline">\(c_0\)</span> and degrees of freedom from 5 to 7 to gauge the effect of the regularization.</p>
-<div class="sourceCode" id="cb34"><html><body><pre class="r"><span class="no">c0_values</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fl">.5</span>, <span class="fl">1</span>, <span class="fl">2</span>, <span class="fl">4</span>, <span class="fl">8</span>, <span class="fl">16</span>, <span class="fl">32</span>)
-<span class="no">stable</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(
-    <span class="kw">c0</span> <span class="kw">=</span> <span class="no">c0_values</span>,
-    <span class="kw">`DF 5`</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">c0_values</span>, <span class="kw">function</span>(<span class="no">c0</span>) <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">z</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Normal"</span>, <span class="kw">pDegree</span> <span class="kw">=</span> <span class="fl">5</span>, <span class="kw">c0</span> <span class="kw">=</span> <span class="no">c0</span>)$<span class="no">S</span>),
-    <span class="kw">`DF 6`</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">c0_values</span>, <span class="kw">function</span>(<span class="no">c0</span>) <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">z</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Normal"</span>, <span class="kw">pDegree</span> <span class="kw">=</span> <span class="fl">6</span>, <span class="kw">c0</span> <span class="kw">=</span> <span class="no">c0</span>)$<span class="no">S</span>),
-    <span class="kw">`DF 7`</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">c0_values</span>, <span class="kw">function</span>(<span class="no">c0</span>) <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">z</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Normal"</span>, <span class="kw">pDegree</span> <span class="kw">=</span> <span class="fl">7</span>, <span class="kw">c0</span> <span class="kw">=</span> <span class="no">c0</span>)$<span class="no">S</span>)
-)
-<span class="kw pkg">knitr</span><span class="kw ns">::</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html">kable</a></span>(<span class="no">stable</span>)</pre></body></html></div>
-<table class="table">
-<thead><tr class="header">
-<th align="right">c0</th>
-<th align="right">DF.5</th>
-<th align="right">DF.6</th>
-<th align="right">DF.7</th>
-</tr></thead>
-<tbody>
-<tr class="odd">
-<td align="right">0.5</td>
-<td align="right">0.0010381</td>
-<td align="right">0.0013048</td>
-<td align="right">0.0018553</td>
-</tr>
-<tr class="even">
-<td align="right">1.0</td>
-<td align="right">0.0029450</td>
-<td align="right">0.0034142</td>
-<td align="right">0.0042665</td>
-</tr>
-<tr class="odd">
-<td align="right">2.0</td>
-<td align="right">0.0082974</td>
-<td align="right">0.0090723</td>
-<td align="right">0.0102037</td>
-</tr>
-<tr class="even">
-<td align="right">4.0</td>
-<td align="right">0.0236310</td>
-<td align="right">0.0248144</td>
-<td align="right">0.0257369</td>
-</tr>
-<tr class="odd">
-<td align="right">8.0</td>
-<td align="right">0.0685852</td>
-<td align="right">0.0708393</td>
-<td align="right">0.0697542</td>
-</tr>
-<tr class="even">
-<td align="right">16.0</td>
-<td align="right">0.1915350</td>
-<td align="right">0.2039325</td>
-<td align="right">0.2025109</td>
-</tr>
-<tr class="odd">
-<td align="right">32.0</td>
-<td align="right">0.4831292</td>
-<td align="right">0.5385301</td>
-<td align="right">0.5698289</td>
-</tr>
-</tbody>
-</table>
-<p>A DF of 6 or 7 captures the bimodality and a choice of <span class="math inline">\(c_0 &lt; 4\)</span> would avoid excessive penalization.</p>
-</div>
-<div id="binomial-example" class="section level2">
-<h2 class="hasAnchor">
-<a href="#binomial-example" class="anchor"></a>Binomial Example</h2>
-<p>The dataset <code>surg</code> contains data on intestinal surgery on 844 cancer patients. In the study, surgeons removed <em>satellite</em> nodes for later testing. The data consists of pairs <span class="math inline">\((n_i, X_i)\)</span> where <span class="math inline">\(n_i\)</span> is the number of satellites removed and <span class="math inline">\(X_i\)</span> is the number found to be malignant among them.</p>
-<p>We assume a binomial model with <span class="math inline">\(X_i \sim Binomial(n_i, \theta_i)\)</span> with <span class="math inline">\(\theta_i\)</span> being the probability of any one satellite site being malignant for the <span class="math inline">\(i\)</span>th patient.</p>
-<p>We take <span class="math inline">\(\mathcal{T} = (0.01, 0.02,\ldots, 0.09)\)</span>, so <span class="math inline">\(m = 99.\)</span> We take <span class="math inline">\(Q\)</span> to be the default 5-degree natural spline with columns standardized to mean 0 and sum of squares equal to 1. The penalization parameter is set to 1. The figure below shows the estimated prior density of <span class="math inline">\(g(\theta)\)</span>.</p>
-<div class="sourceCode" id="cb35"><html><body><pre class="r"><span class="no">tau</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="kw">from</span> <span class="kw">=</span> <span class="fl">0.01</span>, <span class="kw">to</span> <span class="kw">=</span> <span class="fl">0.99</span>, <span class="kw">by</span> <span class="kw">=</span> <span class="fl">0.01</span>)
-<span class="no">result</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">surg</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Binomial"</span>, <span class="kw">c0</span> <span class="kw">=</span> <span class="fl">1</span>)
-<span class="no">d</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(<span class="no">result</span>$<span class="no">stats</span>)
-<span class="no">indices</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="fl">5</span>, <span class="fl">99</span>, <span class="fl">5</span>)
-<span class="no">errorX</span> <span class="kw">&lt;-</span> <span class="no">tau</span>[<span class="no">indices</span>]
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>() +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="no">d</span>, <span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g</span>)) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_linerange.html">geom_errorbar</a></span>(<span class="kw">data</span> <span class="kw">=</span> <span class="no">d</span>[<span class="no">indices</span>, ],
-                  <span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">ymin</span> <span class="kw">=</span> <span class="no">g</span> - <span class="no">SE.g</span>, <span class="kw">ymax</span> <span class="kw">=</span> <span class="no">g</span> + <span class="no">SE.g</span>),
-                  <span class="kw">width</span> <span class="kw">=</span> <span class="fl">.01</span>, <span class="kw">color</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/paste.html">paste</a></span>(<span class="fu">g</span>(<span class="no">theta</span>), <span class="st">" +/- SE"</span>)))</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-32-1.png" width="700"></p>
-<p>The complete table of estimates and standard errors is also available.</p>
-<div class="sourceCode" id="cb36"><html><body><pre class="r"><span class="kw pkg">knitr</span><span class="kw ns">::</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html">kable</a></span>(<span class="no">d</span>[<span class="no">indices</span>, ], <span class="kw">row.names</span> <span class="kw">=</span> <span class="fl">FALSE</span>)</pre></body></html></div>
-<table class="table">
-<thead><tr class="header">
-<th align="right">theta</th>
-<th align="right">g</th>
-<th align="right">SE.g</th>
-<th align="right">G</th>
-<th align="right">SE.G</th>
-<th align="right">Bias.g</th>
-</tr></thead>
-<tbody>
-<tr class="odd">
-<td align="right">0.05</td>
-<td align="right">0.0463457</td>
-<td align="right">0.0015738</td>
-<td align="right">0.4000686</td>
-<td align="right">0.0187280</td>
-<td align="right">-0.0001386</td>
-</tr>
-<tr class="even">
-<td align="right">0.10</td>
-<td align="right">0.0151816</td>
-<td align="right">0.0014905</td>
-<td align="right">0.5227891</td>
-<td align="right">0.0179368</td>
-<td align="right">0.0005557</td>
-</tr>
-<tr class="odd">
-<td align="right">0.15</td>
-<td align="right">0.0063608</td>
-<td align="right">0.0009530</td>
-<td align="right">0.5679756</td>
-<td align="right">0.0177951</td>
-<td align="right">0.0004028</td>
-</tr>
-<tr class="even">
-<td align="right">0.20</td>
-<td align="right">0.0039081</td>
-<td align="right">0.0006332</td>
-<td align="right">0.5910638</td>
-<td align="right">0.0182757</td>
-<td align="right">0.0002736</td>
-</tr>
-<tr class="odd">
-<td align="right">0.25</td>
-<td align="right">0.0038724</td>
-<td align="right">0.0005167</td>
-<td align="right">0.6098155</td>
-<td align="right">0.0187490</td>
-<td align="right">0.0002012</td>
-</tr>
-<tr class="even">
-<td align="right">0.30</td>
-<td align="right">0.0053573</td>
-<td align="right">0.0005917</td>
-<td align="right">0.6330463</td>
-<td align="right">0.0188231</td>
-<td align="right">0.0000993</td>
-</tr>
-<tr class="odd">
-<td align="right">0.35</td>
-<td align="right">0.0082256</td>
-<td align="right">0.0010482</td>
-<td align="right">0.6679794</td>
-<td align="right">0.0180358</td>
-<td align="right">-0.0001306</td>
-</tr>
-<tr class="even">
-<td align="right">0.40</td>
-<td align="right">0.0111402</td>
-<td align="right">0.0015163</td>
-<td align="right">0.7184355</td>
-<td align="right">0.0170109</td>
-<td align="right">-0.0004107</td>
-</tr>
-<tr class="odd">
-<td align="right">0.45</td>
-<td align="right">0.0111343</td>
-<td align="right">0.0011809</td>
-<td align="right">0.7755223</td>
-<td align="right">0.0167801</td>
-<td align="right">-0.0003743</td>
-</tr>
-<tr class="even">
-<td align="right">0.50</td>
-<td align="right">0.0086588</td>
-<td align="right">0.0009940</td>
-<td align="right">0.8242825</td>
-<td align="right">0.0155204</td>
-<td align="right">-0.0000980</td>
-</tr>
-<tr class="odd">
-<td align="right">0.55</td>
-<td align="right">0.0058962</td>
-<td align="right">0.0010407</td>
-<td align="right">0.8590792</td>
-<td align="right">0.0135817</td>
-<td align="right">0.0001197</td>
-</tr>
-<tr class="even">
-<td align="right">0.60</td>
-<td align="right">0.0039564</td>
-<td align="right">0.0008251</td>
-<td align="right">0.8823476</td>
-<td align="right">0.0125338</td>
-<td align="right">0.0002037</td>
-</tr>
-<tr class="odd">
-<td align="right">0.65</td>
-<td align="right">0.0028839</td>
-<td align="right">0.0005341</td>
-<td align="right">0.8986379</td>
-<td align="right">0.0120984</td>
-<td align="right">0.0002143</td>
-</tr>
-<tr class="even">
-<td align="right">0.70</td>
-<td align="right">0.0023398</td>
-<td align="right">0.0004140</td>
-<td align="right">0.9112647</td>
-<td align="right">0.0114461</td>
-<td align="right">0.0002039</td>
-</tr>
-<tr class="odd">
-<td align="right">0.75</td>
-<td align="right">0.0021301</td>
-<td align="right">0.0004763</td>
-<td align="right">0.9222208</td>
-<td align="right">0.0103935</td>
-<td align="right">0.0001911</td>
-</tr>
-<tr class="even">
-<td align="right">0.80</td>
-<td align="right">0.0021935</td>
-<td align="right">0.0005645</td>
-<td align="right">0.9329514</td>
-<td align="right">0.0092023</td>
-<td align="right">0.0001796</td>
-</tr>
-<tr class="odd">
-<td align="right">0.85</td>
-<td align="right">0.0025512</td>
-<td align="right">0.0005742</td>
-<td align="right">0.9448671</td>
-<td align="right">0.0082261</td>
-<td align="right">0.0001632</td>
-</tr>
-<tr class="even">
-<td align="right">0.90</td>
-<td align="right">0.0032568</td>
-<td align="right">0.0005083</td>
-<td align="right">0.9595831</td>
-<td align="right">0.0072551</td>
-<td align="right">0.0001253</td>
-</tr>
-<tr class="odd">
-<td align="right">0.95</td>
-<td align="right">0.0044141</td>
-<td align="right">0.0007505</td>
-<td align="right">0.9791329</td>
-<td align="right">0.0048537</td>
-<td align="right">0.0000355</td>
-</tr>
-</tbody>
-</table>
-<p>The empirical Bayes estimate of the prior distribution puts most of its mass on the small values as can be seem below.</p>
-<div class="sourceCode" id="cb37"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/cat.html">cat</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/sprintf.html">sprintf</a></span>(<span class="st">"Mass below .20 = %0.2f\n"</span>, <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">d</span>[<span class="fl">1</span>:<span class="fl">20</span>, <span class="st">"g"</span>])))</pre></body></html></div>
-<pre><code>## Mass below .20 = 0.59</code></pre>
-<div class="sourceCode" id="cb39"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/cat.html">cat</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/sprintf.html">sprintf</a></span>(<span class="st">"Mass above .80 = %0.2f\n"</span>, <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">d</span>[<span class="fl">80</span>:<span class="fl">99</span>, <span class="st">"g"</span>])))</pre></body></html></div>
-<pre><code>## Mass above .80 = 0.07</code></pre>
-<div id="posterior-estimates" class="section level3">
-<h3 class="hasAnchor">
-<a href="#posterior-estimates" class="anchor"></a>Posterior Estimates</h3>
-<p>The posterior distribution of <span class="math inline">\(\theta_i\)</span> given <span class="math inline">\((n_i, X_i)\)</span> is computed using Bayes rule as</p>
-<p><span class="math display">\[
-\hat{g} (\theta | X_i = x_i, n_i) = 
-    \frac{g_{\hat{\alpha}} (\theta) {n_i \choose x_i} 
-    \theta^{x_i} (1 - \theta)^{n_i-x_i}} {f_{\hat{\alpha}}(n_i, x_i)}
-\]</span></p>
-<p>where the denominator is given by</p>
-<p><span class="math display">\[
-f_\alpha(n_i, x_i) = \int_0^1{n_i \choose x_i}
-    \theta^{x_i}(1-\theta)^{n_i-x_i}g_\alpha(\theta)\,d\theta.
-\]</span></p>
-<p>with the mle <span class="math inline">\(\hat{\alpha}\)</span> in place of <span class="math inline">\(\alpha\)</span>.</p>
-<p>Since <span class="math inline">\(g(\theta)\)</span> is discrete, the integrals are mere sums as shown below.</p>
-<div class="sourceCode" id="cb41"><html><body><pre class="r"><span class="no">theta</span> <span class="kw">&lt;-</span> <span class="no">result</span>$<span class="no">stats</span>[, <span class="st">'theta'</span>]
-<span class="no">gTheta</span> <span class="kw">&lt;-</span> <span class="no">result</span>$<span class="no">stats</span>[, <span class="st">'g'</span>]
-
-<span class="no">f_alpha</span> <span class="kw">&lt;-</span> <span class="kw">function</span>(<span class="no">n_k</span>, <span class="no">x_k</span>) {
-    <span class="co">## .01 is the delta_theta in the Riemann sum</span>
-    <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="fu"><a href="https://rdrr.io/r/stats/Binomial.html">dbinom</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">x_k</span>, <span class="kw">size</span> <span class="kw">=</span> <span class="no">n_k</span>, <span class="kw">prob</span> <span class="kw">=</span> <span class="no">theta</span>) * <span class="no">gTheta</span>) * <span class="fl">.01</span>
-}
-
-<span class="no">g_theta_hat</span> <span class="kw">&lt;-</span> <span class="kw">function</span>(<span class="no">n_k</span>, <span class="no">x_k</span>) {
-    <span class="no">gTheta</span> * <span class="fu"><a href="https://rdrr.io/r/stats/Binomial.html">dbinom</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">x_k</span>, <span class="kw">size</span> <span class="kw">=</span> <span class="no">n_k</span>, <span class="kw">prob</span> <span class="kw">=</span> <span class="no">theta</span>) / <span class="fu">f_alpha</span>(<span class="no">n_k</span>, <span class="no">x_k</span>)
-}</pre></body></html></div>
-<p>We plot a few posterior distributions.</p>
-<div class="sourceCode" id="cb42"><html><body><pre class="r"><span class="no">g1</span> <span class="kw">&lt;-</span> <span class="fu">g_theta_hat</span>(<span class="kw">x_k</span> <span class="kw">=</span> <span class="fl">7</span>, <span class="kw">n_k</span> <span class="kw">=</span> <span class="fl">32</span>)
-<span class="no">g2</span> <span class="kw">&lt;-</span> <span class="fu">g_theta_hat</span>(<span class="kw">x_k</span> <span class="kw">=</span> <span class="fl">3</span>, <span class="kw">n_k</span> <span class="kw">=</span> <span class="fl">6</span>)
-<span class="no">g3</span> <span class="kw">&lt;-</span> <span class="fu">g_theta_hat</span>(<span class="kw">x_k</span> <span class="kw">=</span> <span class="fl">17</span>, <span class="kw">n_k</span> <span class="kw">=</span> <span class="fl">18</span>)
-
-<span class="fu"><a href="https://ggplot2.tidyverse.org/reference/ggplot.html">ggplot</a></span>() +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g1</span>), <span class="kw">col</span> <span class="kw">=</span> <span class="st">"magenta"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/lims.html">ylim</a></span>(<span class="fl">0</span>, <span class="fl">10</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g2</span>), <span class="kw">col</span> <span class="kw">=</span> <span class="st">"red"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/geom_path.html">geom_line</a></span>(<span class="kw">mapping</span> <span class="kw">=</span> <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/aes.html">aes</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="no">theta</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="no">g3</span>), <span class="kw">col</span> <span class="kw">=</span> <span class="st">"blue"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/labs.html">labs</a></span>(<span class="kw">x</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="no">theta</span>), <span class="kw">y</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/expression.html">expression</a></span>(<span class="fu">g</span>(<span class="fu"><a href="https://rdrr.io/r/base/paste.html">paste</a></span>(<span class="no">theta</span>, <span class="st">"|(x, n)"</span>)))) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> <span class="fl">0.15</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fl">4.25</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"x=7, n=32"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> <span class="fl">0.425</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fl">4.25</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"x=3, n=6"</span>) +
-    <span class="fu"><a href="https://ggplot2.tidyverse.org/reference/annotate.html">annotate</a></span>(<span class="st">"text"</span>, <span class="kw">x</span> <span class="kw">=</span> <span class="fl">0.85</span>, <span class="kw">y</span> <span class="kw">=</span> <span class="fl">7.5</span>, <span class="kw">label</span> <span class="kw">=</span> <span class="st">"x=17, n=18"</span>)</pre></body></html></div>
-<p><img src="deconvolution_files/figure-html/unnamed-chunk-36-1.png" width="700"></p>
-<p>The empirical Bayes posterior estimate <span class="math inline">\(\theta^{EB}\)</span> for patients 34, 40, and 679 in the dataset for example is</p>
-<div class="sourceCode" id="cb43"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/cat.html">cat</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/sprintf.html">sprintf</a></span>(<span class="st">"Empirical Bayes Estimate: %f\n"</span>, <span class="fl">0.01</span> * <span class="fu"><a href="https://rdrr.io/r/base/sum.html">sum</a></span>(<span class="no">theta</span> * <span class="no">g2</span>)))</pre></body></html></div>
-<pre><code>## Empirical Bayes Estimate: 0.453516</code></pre>
-</div>
-<div id="bootstrap-comparison-1" class="section level3">
-<h3 class="hasAnchor">
-<a href="#bootstrap-comparison-1" class="anchor"></a>Bootstrap Comparison</h3>
-<p>As a check on the estimates of standard error and bias provided by <code>deconv</code>, we compare the results with what we obtain using a parametric boostrap.</p>
-<p>The boostrap is run as follows. For each of 1000 runs, 844 simulated realizations <span class="math inline">\(\hat{\Theta}^*\)</span> are sampled from density <span class="math inline">\(\hat{g}.\)</span> Each gave an <span class="math inline">\(X_i^* \sim Binomial(n_i, \Theta_i^*)\)</span> with <span class="math inline">\(n_i\)</span> the <span class="math inline">\(i\)</span>th sample in the original data set. Finally, <span class="math inline">\(\hat{\alpha}\)</span> was computed using <code>deconv</code>.</p>
-<div class="sourceCode" id="cb45"><html><body><pre class="r"><span class="fu"><a href="https://rdrr.io/r/base/Random.html">set.seed</a></span>(<span class="fl">32776</span>)
-<span class="no">B</span> <span class="kw">&lt;-</span> <span class="fl">1000</span>
-<span class="no">gHat</span> <span class="kw">&lt;-</span> <span class="no">d</span>$<span class="no">g</span>
-<span class="no">N</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/nrow.html">nrow</a></span>(<span class="no">surg</span>)
-
-<span class="no">genBootSample</span> <span class="kw">&lt;-</span> <span class="kw">function</span>() {
-    <span class="no">thetaStar</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/sample.html">sample</a></span>(<span class="no">tau</span>, <span class="kw">size</span> <span class="kw">=</span> <span class="no">N</span>, <span class="kw">replace</span> <span class="kw">=</span> <span class="fl">TRUE</span>, <span class="kw">prob</span> <span class="kw">=</span> <span class="no">gHat</span>)
-    <span class="no">sStar</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq_len</a></span>(<span class="no">N</span>),
-                    <span class="kw">function</span>(<span class="no">i</span>)
-                        <span class="fu"><a href="https://rdrr.io/r/stats/Binomial.html">rbinom</a></span>(<span class="kw">n</span> <span class="kw">=</span> <span class="fl">1</span> , <span class="kw">size</span> <span class="kw">=</span> <span class="no">surg</span>$<span class="no">n</span>[<span class="no">i</span>], <span class="kw">prob</span> <span class="kw">=</span> <span class="no">thetaStar</span>[<span class="no">i</span>]))
-    <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(<span class="kw">n</span> <span class="kw">=</span> <span class="no">surg</span>$<span class="no">n</span>, <span class="kw">s</span> <span class="kw">=</span> <span class="no">sStar</span>)
-}
-
-<span class="no">bootResults</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">lapply</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq_len</a></span>(<span class="no">B</span>),
-                      <span class="kw">function</span>(<span class="no">k</span>) {
-                          <span class="no">surgBoot</span> <span class="kw">&lt;-</span> <span class="fu">genBootSample</span>()
-                          <span class="no">mat</span> <span class="kw">&lt;-</span> <span class="fu"><a href="../reference/deconv.html">deconv</a></span>(<span class="kw">tau</span> <span class="kw">=</span> <span class="no">tau</span>, <span class="kw">X</span> <span class="kw">=</span> <span class="no">surgBoot</span>, <span class="kw">family</span> <span class="kw">=</span> <span class="st">"Binomial"</span>,
-                                        <span class="kw">c0</span> <span class="kw">=</span> <span class="fl">1</span>)$<span class="no">stats</span>
-                          <span class="no">mat</span>[, <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="st">"g"</span>, <span class="st">"Bias.g"</span>)]
-                      })
-
-<span class="no">gBoot</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">bootResults</span>, <span class="kw">function</span>(<span class="no">x</span>) <span class="no">x</span>[, <span class="fl">1</span>])
-<span class="no">BiasBoot</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/lapply.html">sapply</a></span>(<span class="no">bootResults</span>, <span class="kw">function</span>(<span class="no">x</span>) <span class="no">x</span>[, <span class="fl">2</span>])
-
-<span class="no">indices</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/c.html">c</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/seq.html">seq</a></span>(<span class="fl">1</span>, <span class="fl">99</span>, <span class="fl">11</span>), <span class="fl">99</span>)
-
-<span class="no">table2</span> <span class="kw">&lt;-</span> <span class="fu"><a href="https://rdrr.io/r/base/data.frame.html">data.frame</a></span>(<span class="kw">theta</span> <span class="kw">=</span> <span class="no">tau</span>,
-                     <span class="kw">gTheta</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Round.html">round</a></span>(<span class="no">gHat</span> * <span class="fl">100</span>, <span class="fl">3</span>),
-                     <span class="kw">sdFormula</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Round.html">round</a></span>(<span class="no">d</span>$<span class="no">SE.g</span> * <span class="fl">100</span>, <span class="fl">3</span>),
-                     <span class="kw">sdSimul</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Round.html">round</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">gBoot</span>, <span class="fl">1</span>, <span class="no">sd</span>) * <span class="fl">100</span>, <span class="fl">3</span>),
-                     <span class="kw">BiasFormula</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Round.html">round</a></span>(<span class="no">d</span>$<span class="no">Bias.g</span> * <span class="fl">100</span>, <span class="fl">3</span>),
-                     <span class="kw">BiasSimul</span> <span class="kw">=</span> <span class="fu"><a href="https://rdrr.io/r/base/Round.html">round</a></span>(<span class="fu"><a href="https://rdrr.io/r/base/apply.html">apply</a></span>(<span class="no">BiasBoot</span>, <span class="fl">1</span>, <span class="no">mean</span>) * <span class="fl">100</span>, <span class="fl">3</span>))[ <span class="no">indices</span>, ]</pre></body></html></div>
-<p>We print out some estimated quantities for comparison.</p>
-<div class="sourceCode" id="cb46"><html><body><pre class="r"><span class="kw pkg">knitr</span><span class="kw ns">::</span><span class="fu"><a href="https://rdrr.io/pkg/knitr/man/kable.html">kable</a></span>(<span class="no">table2</span>, <span class="kw">row.names</span> <span class="kw">=</span> <span class="fl">FALSE</span>)</pre></body></html></div>
-<table class="table">
-<thead><tr class="header">
-<th align="right">theta</th>
-<th align="right">gTheta</th>
-<th align="right">sdFormula</th>
-<th align="right">sdSimul</th>
-<th align="right">BiasFormula</th>
-<th align="right">BiasSimul</th>
-</tr></thead>
-<tbody>
-<tr class="odd">
-<td align="right">0.01</td>
-<td align="right">12.326</td>
-<td align="right">0.870</td>
-<td align="right">0.911</td>
-<td align="right">-0.482</td>
-<td align="right">-0.543</td>
-</tr>
-<tr class="even">
-<td align="right">0.12</td>
-<td align="right">1.033</td>
-<td align="right">0.127</td>
-<td align="right">0.135</td>
-<td align="right">0.051</td>
-<td align="right">0.058</td>
-</tr>
-<tr class="odd">
-<td align="right">0.23</td>
-<td align="right">0.369</td>
-<td align="right">0.054</td>
-<td align="right">0.061</td>
-<td align="right">0.023</td>
-<td align="right">0.027</td>
-</tr>
-<tr class="even">
-<td align="right">0.34</td>
-<td align="right">0.757</td>
-<td align="right">0.093</td>
-<td align="right">0.091</td>
-<td align="right">-0.007</td>
-<td align="right">-0.008</td>
-</tr>
-<tr class="odd">
-<td align="right">0.45</td>
-<td align="right">1.113</td>
-<td align="right">0.118</td>
-<td align="right">0.113</td>
-<td align="right">-0.037</td>
-<td align="right">-0.036</td>
-</tr>
-<tr class="even">
-<td align="right">0.56</td>
-<td align="right">0.543</td>
-<td align="right">0.102</td>
-<td align="right">0.097</td>
-<td align="right">0.015</td>
-<td align="right">0.016</td>
-</tr>
-<tr class="odd">
-<td align="right">0.67</td>
-<td align="right">0.262</td>
-<td align="right">0.046</td>
-<td align="right">0.049</td>
-<td align="right">0.021</td>
-<td align="right">0.024</td>
-</tr>
-<tr class="even">
-<td align="right">0.78</td>
-<td align="right">0.213</td>
-<td align="right">0.053</td>
-<td align="right">0.050</td>
-<td align="right">0.018</td>
-<td align="right">0.018</td>
-</tr>
-<tr class="odd">
-<td align="right">0.89</td>
-<td align="right">0.308</td>
-<td align="right">0.052</td>
-<td align="right">0.046</td>
-<td align="right">0.014</td>
-<td align="right">0.013</td>
-</tr>
-<tr class="even">
-<td align="right">0.99</td>
-<td align="right">0.575</td>
-<td align="right">0.158</td>
-<td align="right">0.157</td>
-<td align="right">-0.010</td>
-<td align="right">-0.014</td>
-</tr>
-</tbody>
-</table>
-</div>
-</div>
-  </div>
-
-  <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
-
-        <nav id="toc" data-toggle="toc"><h2 data-toc-skip>Contents</h2>
-    </nav>
-</div>
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-      <footer><div class="copyright">
-  <p>Developed by <a href="http://statweb.stanford.edu/~ckirby/brad/">Bradley Efron</a>, <a href="https://web.stanford.edu/~naras">Balasubramanian Narasimhan</a>.</p>
-</div>
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-  <p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.5.1.</p>
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-
-    <p>Narasimhan B, Efron B (2020).
-&ldquo;deconvolveR: A $G$-Modeling Program for Deconvolution and Empirical Bayes Estimation.&rdquo;
-<em>Journal of Statistical Software</em>, <b>94</b>(11), 1&ndash;20.
-doi: <a href="https://doi.org/10.18637/jss.v094.i11">10.18637/jss.v094.i11</a>. 
-</p>
-    <pre>@Article{,
-  title = {{deconvolveR}: A $G$-Modeling Program for Deconvolution and Empirical Bayes Estimation},
-  author = {Balasubramanian Narasimhan and Bradley Efron},
-  journal = {Journal of Statistical Software},
-  year = {2020},
-  volume = {94},
-  number = {11},
-  pages = {1--20},
-  doi = {10.18637/jss.v094.i11},
-}</pre>
-
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-    </div>
-
-    <ul class="list-unstyled">
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-        <p><strong><a href='http://statweb.stanford.edu/~ckirby/brad/'>Bradley Efron</a></strong>. Author. 
-        </p>
-      </li>
-      <li>
-        <p><strong><a href='https://web.stanford.edu/~naras'>Balasubramanian Narasimhan</a></strong>. Author, maintainer. 
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-nav[data-toggle='toc'] .nav > .active:focus > a {
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-/* Nav: second level (shown on .active) */
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-  padding-bottom: 10px;
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-  font-size: 12px;
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-nav[data-toggle='toc'] .nav .nav > .active:focus > a {
-  padding-left: 28px;
-  font-weight: 500;
-}
-
-/* from https://github.com/twbs/bootstrap/blob/e38f066d8c203c3e032da0ff23cd2d6098ee2dd6/docs/assets/css/src/docs.css#L631-L634 */
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diff --git a/docs/bootstrap-toc.js b/docs/bootstrap-toc.js
deleted file mode 100644
index 1cdd573..0000000
--- a/docs/bootstrap-toc.js
+++ /dev/null
@@ -1,159 +0,0 @@
-/*!
- * Bootstrap Table of Contents v0.4.1 (http://afeld.github.io/bootstrap-toc/)
- * Copyright 2015 Aidan Feldman
- * Licensed under MIT (https://github.com/afeld/bootstrap-toc/blob/gh-pages/LICENSE.md) */
-(function() {
-  'use strict';
-
-  window.Toc = {
-    helpers: {
-      // return all matching elements in the set, or their descendants
-      findOrFilter: function($el, selector) {
-        // http://danielnouri.org/notes/2011/03/14/a-jquery-find-that-also-finds-the-root-element/
-        // http://stackoverflow.com/a/12731439/358804
-        var $descendants = $el.find(selector);
-        return $el.filter(selector).add($descendants).filter(':not([data-toc-skip])');
-      },
-
-      generateUniqueIdBase: function(el) {
-        var text = $(el).text();
-        var anchor = text.trim().toLowerCase().replace(/[^A-Za-z0-9]+/g, '-');
-        return anchor || el.tagName.toLowerCase();
-      },
-
-      generateUniqueId: function(el) {
-        var anchorBase = this.generateUniqueIdBase(el);
-        for (var i = 0; ; i++) {
-          var anchor = anchorBase;
-          if (i > 0) {
-            // add suffix
-            anchor += '-' + i;
-          }
-          // check if ID already exists
-          if (!document.getElementById(anchor)) {
-            return anchor;
-          }
-        }
-      },
-
-      generateAnchor: function(el) {
-        if (el.id) {
-          return el.id;
-        } else {
-          var anchor = this.generateUniqueId(el);
-          el.id = anchor;
-          return anchor;
-        }
-      },
-
-      createNavList: function() {
-        return $('<ul class="nav"></ul>');
-      },
-
-      createChildNavList: function($parent) {
-        var $childList = this.createNavList();
-        $parent.append($childList);
-        return $childList;
-      },
-
-      generateNavEl: function(anchor, text) {
-        var $a = $('<a></a>');
-        $a.attr('href', '#' + anchor);
-        $a.text(text);
-        var $li = $('<li></li>');
-        $li.append($a);
-        return $li;
-      },
-
-      generateNavItem: function(headingEl) {
-        var anchor = this.generateAnchor(headingEl);
-        var $heading = $(headingEl);
-        var text = $heading.data('toc-text') || $heading.text();
-        return this.generateNavEl(anchor, text);
-      },
-
-      // Find the first heading level (`<h1>`, then `<h2>`, etc.) that has more than one element. Defaults to 1 (for `<h1>`).
-      getTopLevel: function($scope) {
-        for (var i = 1; i <= 6; i++) {
-          var $headings = this.findOrFilter($scope, 'h' + i);
-          if ($headings.length > 1) {
-            return i;
-          }
-        }
-
-        return 1;
-      },
-
-      // returns the elements for the top level, and the next below it
-      getHeadings: function($scope, topLevel) {
-        var topSelector = 'h' + topLevel;
-
-        var secondaryLevel = topLevel + 1;
-        var secondarySelector = 'h' + secondaryLevel;
-
-        return this.findOrFilter($scope, topSelector + ',' + secondarySelector);
-      },
-
-      getNavLevel: function(el) {
-        return parseInt(el.tagName.charAt(1), 10);
-      },
-
-      populateNav: function($topContext, topLevel, $headings) {
-        var $context = $topContext;
-        var $prevNav;
-
-        var helpers = this;
-        $headings.each(function(i, el) {
-          var $newNav = helpers.generateNavItem(el);
-          var navLevel = helpers.getNavLevel(el);
-
-          // determine the proper $context
-          if (navLevel === topLevel) {
-            // use top level
-            $context = $topContext;
-          } else if ($prevNav && $context === $topContext) {
-            // create a new level of the tree and switch to it
-            $context = helpers.createChildNavList($prevNav);
-          } // else use the current $context
-
-          $context.append($newNav);
-
-          $prevNav = $newNav;
-        });
-      },
-
-      parseOps: function(arg) {
-        var opts;
-        if (arg.jquery) {
-          opts = {
-            $nav: arg
-          };
-        } else {
-          opts = arg;
-        }
-        opts.$scope = opts.$scope || $(document.body);
-        return opts;
-      }
-    },
-
-    // accepts a jQuery object, or an options object
-    init: function(opts) {
-      opts = this.helpers.parseOps(opts);
-
-      // ensure that the data attribute is in place for styling
-      opts.$nav.attr('data-toggle', 'toc');
-
-      var $topContext = this.helpers.createChildNavList(opts.$nav);
-      var topLevel = this.helpers.getTopLevel(opts.$scope);
-      var $headings = this.helpers.getHeadings(opts.$scope, topLevel);
-      this.helpers.populateNav($topContext, topLevel, $headings);
-    }
-  };
-
-  $(function() {
-    $('nav[data-toggle="toc"]').each(function(i, el) {
-      var $nav = $(el);
-      Toc.init($nav);
-    });
-  });
-})();
diff --git a/docs/docsearch.css b/docs/docsearch.css
deleted file mode 100644
index e5f1fe1..0000000
--- a/docs/docsearch.css
+++ /dev/null
@@ -1,148 +0,0 @@
-/* Docsearch -------------------------------------------------------------- */
-/*
-  Source: https://github.com/algolia/docsearch/
-  License: MIT
-*/
-
-.algolia-autocomplete {
-  display: block;
-  -webkit-box-flex: 1;
-  -ms-flex: 1;
-  flex: 1
-}
-
-.algolia-autocomplete .ds-dropdown-menu {
-  width: 100%;
-  min-width: none;
-  max-width: none;
-  padding: .75rem 0;
-  background-color: #fff;
-  background-clip: padding-box;
-  border: 1px solid rgba(0, 0, 0, .1);
-  box-shadow: 0 .5rem 1rem rgba(0, 0, 0, .175);
-}
-
-@media (min-width:768px) {
-  .algolia-autocomplete .ds-dropdown-menu {
-      width: 175%
-  }
-}
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-.algolia-autocomplete .ds-dropdown-menu::before {
-  display: none
-}
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-.algolia-autocomplete .ds-dropdown-menu [class^=ds-dataset-] {
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-  background-color: rgb(255,255,255);
-  border: 0;
-  max-height: 80vh;
-}
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-.algolia-autocomplete .ds-dropdown-menu .ds-suggestions {
-  margin-top: 0
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-.algolia-autocomplete .algolia-docsearch-suggestion {
-  padding: 0;
-  overflow: visible
-}
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-.algolia-autocomplete .algolia-docsearch-suggestion--category-header {
-  padding: .125rem 1rem;
-  margin-top: 0;
-  font-size: 1.3em;
-  font-weight: 500;
-  color: #00008B;
-  border-bottom: 0
-}
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-.algolia-autocomplete .algolia-docsearch-suggestion--wrapper {
-    float: none;
-    padding-top: 0
-}
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-.algolia-autocomplete .algolia-docsearch-suggestion--subcategory-column {
-  float: none;
-  width: auto;
-  padding: 0;
-  text-align: left
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-.algolia-autocomplete .algolia-docsearch-suggestion--content {
-  float: none;
-  width: auto;
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-.algolia-autocomplete .algolia-docsearch-suggestion--content::before {
-  display: none
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-.algolia-autocomplete .ds-suggestion:not(:first-child) .algolia-docsearch-suggestion--category-header {
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-  border-top: 1px solid rgba(0, 0, 0, .1)
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-  display: block;
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-  margin-bottom: 0.1;
-  font-size: 1.0em;
-  font-weight: 400
-  /* display: none */
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-.algolia-autocomplete .algolia-docsearch-suggestion--title {
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-  padding: .25rem 1rem;
-  margin-bottom: 0;
-  font-size: 0.9em;
-  font-weight: 400
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-.algolia-autocomplete .algolia-docsearch-suggestion--text {
-  padding: 0 1rem .5rem;
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-  font-size: 0.8em;
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-  line-height: 1.25
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-
-.algolia-autocomplete .algolia-docsearch-footer {
-  width: 110px;
-  height: 20px;
-  z-index: 3;
-  margin-top: 10.66667px;
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-    An unknown prior distribution (g) has yielded (unobservable) parameters, each of
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-<a href="#empirical-bayes-deconvolution" class="anchor"></a>Empirical Bayes Deconvolution</h1></div>
-
-<p>An unknown prior density <span class="math inline"><em>g</em>(<em>θ</em>)</span> has yielded (unobservable) <span class="math inline"><em>Θ</em><sub>1</sub>, <em>Θ</em><sub>2</sub>, …, <em>Θ</em><sub><em>N</em></sub></span>, and each <span class="math inline"><em>Θ</em><sub><em>i</em></sub></span> produces an observation <span class="math inline"><em>X</em><sub><em>i</em></sub></span> from an exponential family. <code>deconvolveR</code> is an R package for estimating prior distribution <span class="math inline"><em>g</em>(<em>θ</em>)</span> from the data using Empirical Bayes deconvolution.</p>
-<p>Details and examples may be found in the paper by <a href="https://dx.doi.org/10.18637/jss.v094.i11">Narasimhan and Efron, 2020</a>. A vignette with further examples is also provided.</p>
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-    <h1>Shakespeare word counts in the entire canon: 14,376 distinct words
-appeared exactly once, 4343 words appeared twice etc.</h1>
-    <small class="dont-index">Source: <a href='https://github.com/bnaras/deconvolveR/blob/master/R/deconvolveR-package.R'><code>R/deconvolveR-package.R</code></a></small>
-    <div class="hidden name"><code>bardWordCount.Rd</code></div>
-    </div>
-
-    <div class="ref-description">
-    <p>Shakespeare word counts in the entire canon: 14,376 distinct words
-appeared exactly once, 4343 words appeared twice etc.</p>
-    </div>
-
-    <pre class="usage"><span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span>(<span class='no'>bardWordCount</span>)</pre>
-
-
-    <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
-
-    <p>Bradley Efron and Ronald Thisted. Estimating the number of
-unseen species: How many words did Shakespeare know? Biometrika, Vol 63(3),
-doi:10.1093/biomet/63.3.435.</p>
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-  <p>Developed by <a href='http://statweb.stanford.edu/~ckirby/brad/'>Bradley Efron</a>, <a href='https://web.stanford.edu/~naras'>Balasubramanian Narasimhan</a>.</p>
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diff --git a/docs/reference/deconv-1.png b/docs/reference/deconv-1.png
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-    <div class="page-header">
-    <h1>A function to compute Empirical Bayes estimates using deconvolution</h1>
-    <small class="dont-index">Source: <a href='https://github.com/bnaras/deconvolveR/blob/master/R/deconv.R'><code>R/deconv.R</code></a></small>
-    <div class="hidden name"><code>deconv.Rd</code></div>
-    </div>
-
-    <div class="ref-description">
-    <p>A function to compute Empirical Bayes estimates using deconvolution</p>
-    </div>
-
-    <pre class="usage"><span class='fu'>deconv</span>(
-  <span class='no'>tau</span>,
-  <span class='no'>X</span>,
-  <span class='no'>y</span>,
-  <span class='no'>Q</span>,
-  <span class='no'>P</span>,
-  <span class='kw'>n</span> <span class='kw'>=</span> <span class='fl'>40</span>,
-  <span class='kw'>family</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span>(<span class='st'>"Poisson"</span>, <span class='st'>"Normal"</span>, <span class='st'>"Binomial"</span>),
-  <span class='kw'>ignoreZero</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
-  <span class='kw'>deltaAt</span> <span class='kw'>=</span> <span class='kw'>NULL</span>,
-  <span class='kw'>c0</span> <span class='kw'>=</span> <span class='fl'>1</span>,
-  <span class='kw'>scale</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>,
-  <span class='kw'>pDegree</span> <span class='kw'>=</span> <span class='fl'>5</span>,
-  <span class='kw'>aStart</span> <span class='kw'>=</span> <span class='fl'>1</span>,
-  <span class='no'>...</span>
-)</pre>
-
-    <h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
-    <table class="ref-arguments">
-    <colgroup><col class="name" /><col class="desc" /></colgroup>
-    <tr>
-      <th>tau</th>
-      <td><p>a vector of (implicitly m) discrete support points for
-\(\theta\). For the Poisson and normal families, \(\theta\)
-is the mean parameter and for the binomial, it is the
-probability of success.</p></td>
-    </tr>
-    <tr>
-      <th>X</th>
-      <td><p>the vector of sample values: a vector of counts for
-Poisson, a vector of z-scores for Normal, a 2-d matrix with
-rows consisting of pairs, (trial size \(n_i\), number of
-successes \(X_i\)) for Binomial. See details below</p></td>
-    </tr>
-    <tr>
-      <th>y</th>
-      <td><p>the multinomial counts. See details below</p></td>
-    </tr>
-    <tr>
-      <th>Q</th>
-      <td><p>the Q matrix, implies y and P are supplied as well; see
-details below</p></td>
-    </tr>
-    <tr>
-      <th>P</th>
-      <td><p>the P matrix, implies Q and y are supplied as well; see
-details below</p></td>
-    </tr>
-    <tr>
-      <th>n</th>
-      <td><p>the number of support points for X. Applies only to
-Poisson and Normal. In the former, implies that support of X is
-1 to n or 0 to n-1 depending on the <code>ignoreZero</code> parameter
-below. In the latter, the range of X is divided into n bins to
-construct the multinomial sufficient statistic y (\(y_k\) =
-number of X in bin K) described in the references below</p></td>
-    </tr>
-    <tr>
-      <th>family</th>
-      <td><p>the exponential family, one of <code><a href='https://rdrr.io/r/base/c.html'>c("Poisson",
-"Normal", "Binomial")</a></code> with <code>"Poisson"</code>, the default</p></td>
-    </tr>
-    <tr>
-      <th>ignoreZero</th>
-      <td><p>if the zero values should be ignored (default =
-<code>TRUE</code>). Applies to Poisson only and has the effect of
-adjusting <code>P</code> for the truncation at zero</p></td>
-    </tr>
-    <tr>
-      <th>deltaAt</th>
-      <td><p>the theta value where a delta function is desired
-(default <code>NULL</code>). This applies to the Normal case only and
-even then only if it is non-null.</p></td>
-    </tr>
-    <tr>
-      <th>c0</th>
-      <td><p>the regularization parameter (default 1)</p></td>
-    </tr>
-    <tr>
-      <th>scale</th>
-      <td><p>if the Q matrix should be scaled so that the spline
-basis has mean 0 and columns sum of squares to be one, (default
-<code>TRUE</code>)</p></td>
-    </tr>
-    <tr>
-      <th>pDegree</th>
-      <td><p>the degree of the splines to use (default 5). In
-notation used in the references below, \(p\) = pDegree + 1</p></td>
-    </tr>
-    <tr>
-      <th>aStart</th>
-      <td><p>the starting values for the non-linear optimization,
-default is a vector of 1s</p></td>
-    </tr>
-    <tr>
-      <th>...</th>
-      <td><p>further args to function <code>nlm</code></p></td>
-    </tr>
-    </table>
-
-    <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
-
-    <p>a list of 9 items consisting of</p>
-<dt>mle</dt><dd><p>the maximum
-    likelihood estimate \(\hat{\alpha}\)</p></dd> <dt>Q</dt><dd><p>the m by p
-    matrix Q</p></dd> <dt>P</dt><dd><p>the n by m matrix P</p></dd> <dt>S</dt><dd><p>the ratio of
-    artificial to genuine information per the reference below,
-    where it was referred to as \(R(\alpha)\)</p></dd> <dt>cov</dt><dd><p>the
-    covariance matrix for the mle</p></dd> <dt>cov.g</dt><dd><p>the covariance
-    matrix for the \(g\)</p></dd> <dt>stats</dt><dd><p>an m by 6 or 7 matrix
-    containing columns for \(theta\), \(g\), \(\tilde{g}\)
-    which is \(g\) with thinning correction applied and named
-    <code>tg</code>, std. error of \(g\), \(G\) (the cdf of g),
-    std. error of \(G\), and the bias of \(g\)</p></dd>
-    <dt>loglik</dt><dd><p>the negative log-likelihood function for the data
-    taking a \(p\)-vector argument</p></dd> <dt>statsFunction</dt><dd><p>a
-    function to compute the statistics returned above</p></dd>
-
-    <h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>
-
-    
-
-<p>The data <code>X</code> is always required with two exceptions. In the Poisson case,
-<code>y</code> alone may be specified and <code>X</code> omitted, in which case the sample space of
-the observations $\(X\)$ is assumed to be 1, 2, .., <code><a href='https://rdrr.io/r/base/length.html'>length(y)</a></code>. The second exception is
-for experimentation with other exponential families besides the three implemented here:
-<code>y</code>, <code>P</code> and <code>Q</code> can be specified together.</p>
-<p>Note also that in the Poisson case where there is zero truncation,
-the <code>stats</code> matrix has an additional column <code>"tg"</code> which
-accounts for the thinning correction induced by the truncation. See
-vignette for details.</p>
-    <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
-
-    <p>Bradley Efron. Empirical Bayes Deconvolution Estimates. Biometrika 103(1), 1-20,
-ISSN 0006-3444. doi:10.1093/biomet/asv068.
-<a href='http://biomet.oxfordjournals.org/content/103/1/1.full.pdf+html'>http://biomet.oxfordjournals.org/content/103/1/1.full.pdf+html</a></p>
-<p>Bradley Efron and Trevor Hastie. Computer Age Statistical Inference.
-Cambridge University Press. ISBN 978-1-1-7-14989-2. Chapter 21.</p>
-
-    <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
-    <pre class="examples"><div class='input'><span class='fu'><a href='https://rdrr.io/r/base/Random.html'>set.seed</a></span>(<span class='fl'>238923</span>) <span class='co'>## for reproducibility</span>
-<span class='no'>N</span> <span class='kw'>&lt;-</span> <span class='fl'>1000</span>
-<span class='no'>theta</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/stats/Chisquare.html'>rchisq</a></span>(<span class='no'>N</span>,  <span class='kw'>df</span> <span class='kw'>=</span> <span class='fl'>10</span>)
-<span class='no'>X</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/stats/Poisson.html'>rpois</a></span>(<span class='kw'>n</span> <span class='kw'>=</span> <span class='no'>N</span>, <span class='kw'>lambda</span> <span class='kw'>=</span> <span class='no'>theta</span>)
-<span class='no'>tau</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='fl'>1</span>, <span class='fl'>32</span>)
-<span class='no'>result</span> <span class='kw'>&lt;-</span> <span class='fu'>deconv</span>(<span class='kw'>tau</span> <span class='kw'>=</span> <span class='no'>tau</span>, <span class='kw'>X</span> <span class='kw'>=</span> <span class='no'>X</span>, <span class='kw'>ignoreZero</span> <span class='kw'>=</span> <span class='fl'>FALSE</span>)
-<span class='fu'><a href='https://rdrr.io/r/base/print.html'>print</a></span>(<span class='no'>result</span>$<span class='no'>stats</span>)</div><div class='output co'>#&gt;       theta            g         SE.g           G         SE.G        Bias.g
-#&gt;  [1,]     1 0.0051481615 0.0016525341 0.005148161 1.652534e-03  8.541487e-04
-#&gt;  [2,]     2 0.0098958954 0.0023381444 0.015044057 3.986483e-03  1.099087e-03
-#&gt;  [3,]     3 0.0185198134 0.0029466370 0.033563870 6.895555e-03  1.102607e-03
-#&gt;  [4,]     4 0.0328530174 0.0032249075 0.066416888 9.873512e-03  4.812427e-04
-#&gt;  [5,]     5 0.0537835715 0.0036693678 0.120200459 1.224156e-02 -1.090720e-03
-#&gt;  [6,]     6 0.0791112606 0.0056206577 0.199311720 1.425433e-02 -3.316963e-03
-#&gt;  [7,]     7 0.1017934505 0.0077509950 0.301105170 1.743623e-02 -4.781816e-03
-#&gt;  [8,]     8 0.1121459629 0.0073194510 0.413251133 2.110274e-02 -3.712078e-03
-#&gt;  [9,]     9 0.1076955317 0.0051046274 0.520946665 2.259374e-02 -5.759912e-04
-#&gt; [10,]    10 0.0934081064 0.0049730965 0.614354771 2.125191e-02  2.609017e-03
-#&gt; [11,]    11 0.0758233573 0.0063169145 0.690178128 1.862765e-02  4.426107e-03
-#&gt; [12,]    12 0.0596913949 0.0066510289 0.749869523 1.672495e-02  4.632921e-03
-#&gt; [13,]    13 0.0472247251 0.0058223171 0.797094249 1.624772e-02  3.649927e-03
-#&gt; [14,]    14 0.0388249870 0.0043740118 0.835919236 1.627164e-02  1.933483e-03
-#&gt; [15,]    15 0.0332241328 0.0031665912 0.869143368 1.566917e-02 -3.121847e-05
-#&gt; [16,]    16 0.0289251034 0.0030459927 0.898068472 1.404924e-02 -1.798007e-03
-#&gt; [17,]    17 0.0250253380 0.0036190550 0.923093810 1.171550e-02 -3.024715e-03
-#&gt; [18,]    18 0.0210171384 0.0039132116 0.944110948 9.419157e-03 -3.448033e-03
-#&gt; [19,]    19 0.0167363462 0.0035333178 0.960847294 7.853539e-03 -2.991447e-03
-#&gt; [20,]    20 0.0123508104 0.0026193414 0.973198105 6.965651e-03 -1.879392e-03
-#&gt; [21,]    21 0.0084417110 0.0017508155 0.981639816 6.220128e-03 -6.879592e-04
-#&gt; [22,]    22 0.0054981488 0.0012927504 0.987137965 5.413467e-03  1.279342e-04
-#&gt; [23,]    23 0.0035175978 0.0010712165 0.990655562 4.617829e-03  5.320583e-04
-#&gt; [24,]    24 0.0022788319 0.0008958692 0.992934394 3.919108e-03  6.645066e-04
-#&gt; [25,]    25 0.0015410174 0.0007350941 0.994475412 3.340415e-03  6.679252e-04
-#&gt; [26,]    26 0.0011212384 0.0006048170 0.995596650 2.858930e-03  6.345106e-04
-#&gt; [27,]    27 0.0008930759 0.0005149033 0.996489726 2.434629e-03  6.091289e-04
-#&gt; [28,]    28 0.0007695787 0.0004644742 0.997259305 2.029782e-03  6.030704e-04
-#&gt; [29,]    29 0.0007062477 0.0004516035 0.997965553 1.611586e-03  6.169472e-04
-#&gt; [30,]    30 0.0006794627 0.0004755882 0.998645015 1.150613e-03  6.487035e-04
-#&gt; [31,]    31 0.0006745962 0.0005341491 0.999319611 6.204563e-04  6.948837e-04
-#&gt; [32,]    32 0.0006803885 0.0006204563 1.000000000 1.706969e-10  7.501296e-04</div><div class='input'><span class='co'>##</span>
-<span class='co'>## Twin Towers Example</span>
-<span class='co'>## See Brad Efron: Bayes, Oracle Bayes and Empirical Bayes</span>
-<span class='co'>## disjointTheta is provided by deconvolveR package</span>
-<span class='no'>theta</span> <span class='kw'>&lt;-</span> <span class='no'>disjointTheta</span>; <span class='no'>N</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/length.html'>length</a></span>(<span class='no'>disjointTheta</span>)
-<span class='no'>z</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/stats/Normal.html'>rnorm</a></span>(<span class='kw'>n</span> <span class='kw'>=</span> <span class='no'>N</span>, <span class='kw'>mean</span> <span class='kw'>=</span> <span class='no'>disjointTheta</span>)
-<span class='no'>tau</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/r/base/seq.html'>seq</a></span>(<span class='kw'>from</span> <span class='kw'>=</span> -<span class='fl'>4</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='fl'>5</span>, <span class='kw'>by</span> <span class='kw'>=</span> <span class='fl'>0.2</span>)
-<span class='no'>result</span> <span class='kw'>&lt;-</span> <span class='fu'>deconv</span>(<span class='kw'>tau</span> <span class='kw'>=</span> <span class='no'>tau</span>, <span class='kw'>X</span> <span class='kw'>=</span> <span class='no'>z</span>, <span class='kw'>family</span> <span class='kw'>=</span> <span class='st'>"Normal"</span>, <span class='kw'>pDegree</span> <span class='kw'>=</span> <span class='fl'>6</span>)
-<span class='no'>g</span> <span class='kw'>&lt;-</span> <span class='no'>result</span>$<span class='no'>stats</span>[, <span class='st'>"g"</span>]
-<span class='kw'>if</span> (<span class='fu'><a href='https://rdrr.io/r/base/library.html'>require</a></span>(<span class='st'>"ggplot2"</span>)) {
-  <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/ggplot.html'>ggplot</a></span>() +
-     <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/geom_histogram.html'>geom_histogram</a></span>(<span class='kw'>mapping</span> <span class='kw'>=</span> <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/aes.html'>aes</a></span>(<span class='kw'>x</span> <span class='kw'>=</span> <span class='no'>disjointTheta</span>, <span class='kw'>y</span>  <span class='kw'>=</span> <span class='no'>..count..</span> / <span class='fu'><a href='https://rdrr.io/r/base/sum.html'>sum</a></span>(<span class='no'>..count..</span>) ),
-                    <span class='kw'>color</span> <span class='kw'>=</span> <span class='st'>"blue"</span>, <span class='kw'>fill</span> <span class='kw'>=</span> <span class='st'>"red"</span>, <span class='kw'>bins</span> <span class='kw'>=</span> <span class='fl'>40</span>, <span class='kw'>alpha</span> <span class='kw'>=</span> <span class='fl'>0.5</span>) +
-     <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/geom_histogram.html'>geom_histogram</a></span>(<span class='kw'>mapping</span> <span class='kw'>=</span> <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/aes.html'>aes</a></span>(<span class='kw'>x</span> <span class='kw'>=</span> <span class='no'>z</span>, <span class='kw'>y</span>  <span class='kw'>=</span> <span class='no'>..count..</span> / <span class='fu'><a href='https://rdrr.io/r/base/sum.html'>sum</a></span>(<span class='no'>..count..</span>) ),
-                    <span class='kw'>color</span> <span class='kw'>=</span> <span class='st'>"brown"</span>, <span class='kw'>bins</span> <span class='kw'>=</span> <span class='fl'>40</span>, <span class='kw'>alpha</span> <span class='kw'>=</span> <span class='fl'>0.5</span>) +
-     <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/geom_path.html'>geom_line</a></span>(<span class='kw'>mapping</span> <span class='kw'>=</span> <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/aes.html'>aes</a></span>(<span class='kw'>x</span> <span class='kw'>=</span> <span class='no'>tau</span>, <span class='kw'>y</span> <span class='kw'>=</span> <span class='no'>g</span>), <span class='kw'>color</span> <span class='kw'>=</span> <span class='st'>"black"</span>) +
-     <span class='fu'><a href='https://ggplot2.tidyverse.org/reference/labs.html'>labs</a></span>(<span class='kw'>x</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/paste.html'>paste</a></span>(<span class='fu'><a href='https://rdrr.io/r/base/expression.html'>expression</a></span>(<span class='no'>theta</span>), <span class='st'>"and x"</span>), <span class='kw'>y</span> <span class='kw'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/paste.html'>paste</a></span>(<span class='fu'><a href='https://rdrr.io/r/base/expression.html'>expression</a></span>(<span class='fu'>g</span>(<span class='no'>theta</span>)), <span class='st'>" and f(x)"</span>))
-}</div><div class='output co'>#&gt; <span class='message'>Loading required package: ggplot2</span></div><div class='img'><img src='deconv-1.png' alt='' width='700' height='433' /></div><div class='input'>
-</div></pre>
-  </div>
-  <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
-    <nav id="toc" data-toggle="toc" class="sticky-top">
-      <h2 data-toc-skip>Contents</h2>
-    </nav>
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-      <footer>
-      <div class="copyright">
-  <p>Developed by <a href='http://statweb.stanford.edu/~ckirby/brad/'>Bradley Efron</a>, <a href='https://web.stanford.edu/~naras'>Balasubramanian Narasimhan</a>.</p>
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diff --git a/docs/reference/deconvolveR-package.html b/docs/reference/deconvolveR-package.html
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-containing a number of examples.</p>
-    </div>
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-    <p>Bradley Efron. Empirical Bayes Deconvolution
-    Estimates. Biometrika 103(1), 1-20, ISSN
-    0006-3444. doi:10.1093/biomet/asv068.
-    <a href='http://biomet.oxfordjournals.org/content/103/1/1.full.pdf+html'>http://biomet.oxfordjournals.org/content/103/1/1.full.pdf+html</a></p>
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-    Efron. deconvolveR: A G-Modeling Program for Deconvolution and
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-and Estimation. A friendly introduction is provided in the JSS
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-containing a number of examples.</p>
-    </div>
-
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-    <p>Bradley Efron. Empirical Bayes Deconvolution
-    Estimates. Biometrika 103(1), 1-20, ISSN
-    0006-3444. doi:10.1093/biomet/asv068.
-    <a href='http://biomet.oxfordjournals.org/content/103/1/1.full.pdf+html'>http://biomet.oxfordjournals.org/content/103/1/1.full.pdf+html</a></p>
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-    <h1>A set of \(\Theta\) values that have a bimodal distribution for testing</h1>
-    <small class="dont-index">Source: <a href='https://github.com/bnaras/deconvolveR/blob/master/R/deconvolveR-package.R'><code>R/deconvolveR-package.R</code></a></small>
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-    <div class="ref-description">
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-    </div>
-
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-      
-      <col class="alias" />
-      <col class="title" />
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-
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-      <tr>
-        <th colspan="2">
-          <h2 id="section-all-functions" class="hasAnchor"><a href="#section-all-functions" class="anchor"></a>All functions</h2>
-          <p class="section-desc"></p>
-        </th>
-      </tr>
-      
-      
-    </tbody><tbody>
-      
-      
-      <tr>
-        
-        <td>
-          <p><code><a href="bardWordCount.html">bardWordCount</a></code> </p>
-        </td>
-        <td><p>Shakespeare word counts in the entire canon: 14,376 distinct words
-appeared exactly once, 4343 words appeared twice etc.</p></td>
-      </tr><tr>
-        
-        <td>
-          <p><code><a href="deconv.html">deconv()</a></code> </p>
-        </td>
-        <td><p>A function to compute Empirical Bayes estimates using deconvolution</p></td>
-      </tr><tr>
-        
-        <td>
-          <p><code><a href="deconvolveR-package.html">deconvolveR-package</a></code> </p>
-        </td>
-        <td><p>R package for Empirical Bayes \(g\)-modeling using exponential families.</p></td>
-      </tr><tr>
-        
-        <td>
-          <p><code><a href="disjointTheta.html">disjointTheta</a></code> </p>
-        </td>
-        <td><p>A set of \(\Theta\) values that have a bimodal distribution for testing</p></td>
-      </tr><tr>
-        
-        <td>
-          <p><code><a href="surg.html">surg</a></code> </p>
-        </td>
-        <td><p>Intestinal surgery data involving 844 cancer patients. The data consists of
-pairs (\(n_i\), \(s_i\)) where \(n_i\) is the number of satellites removed
-and \(s_i\) is the number of satellites found to be malignant.</p></td>
-      </tr>
-    </tbody>
-    </table>
-  </div>
-
-  <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
-    <nav id="toc" data-toggle="toc" class="sticky-top">
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-  <p>Developed by <a href='http://statweb.stanford.edu/~ckirby/brad/'>Bradley Efron</a>, <a href='https://web.stanford.edu/~naras'>Balasubramanian Narasimhan</a>.</p>
-</div>
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-  <p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.5.1.</p>
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diff --git a/docs/reference/surg.html b/docs/reference/surg.html
deleted file mode 100644
index f8fba02..0000000
--- a/docs/reference/surg.html
+++ /dev/null
@@ -1,181 +0,0 @@
-<!-- Generated by pkgdown: do not edit by hand -->
-<!DOCTYPE html>
-<html lang="en">
-  <head>
-  <meta charset="utf-8">
-<meta http-equiv="X-UA-Compatible" content="IE=edge">
-<meta name="viewport" content="width=device-width, initial-scale=1.0">
-
-<title>Intestinal surgery data involving 844 cancer patients. The data consists of
-pairs (\(n_i\), \(s_i\)) where \(n_i\) is the number of satellites removed
-and \(s_i\) is the number of satellites found to be malignant. — surg • deconvolveR</title>
-
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-pairs (\(n_i\), \(s_i\)) where \(n_i\) is the number of satellites removed
-and \(s_i\) is the number of satellites found to be malignant. — surg" />
-<meta property="og:description" content="Intestinal surgery data involving 844 cancer patients. The data consists of
-pairs (\(n_i\), \(s_i\)) where \(n_i\) is the number of satellites removed
-and \(s_i\) is the number of satellites found to be malignant." />
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-    <span class="caret"></span>
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-      <a href="../articles/deconvolution.html">Empirical Bayes Deconvolution</a>
-    </li>
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-  <a href="../news/index.html">Changelog</a>
-</li>
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-</div><!--/.navbar -->
-
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-
-<div class="row">
-  <div class="col-md-9 contents">
-    <div class="page-header">
-    <h1>Intestinal surgery data involving 844 cancer patients. The data consists of
-pairs (\(n_i\), \(s_i\)) where \(n_i\) is the number of satellites removed
-and \(s_i\) is the number of satellites found to be malignant.</h1>
-    <small class="dont-index">Source: <a href='https://github.com/bnaras/deconvolveR/blob/master/R/deconvolveR-package.R'><code>R/deconvolveR-package.R</code></a></small>
-    <div class="hidden name"><code>surg.Rd</code></div>
-    </div>
-
-    <div class="ref-description">
-    <p>Intestinal surgery data involving 844 cancer patients. The data consists of
-pairs (\(n_i\), \(s_i\)) where \(n_i\) is the number of satellites removed
-and \(s_i\) is the number of satellites found to be malignant.</p>
-    </div>
-
-    <pre class="usage"><span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span>(<span class='no'>surg</span>)</pre>
-
-
-    <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
-
-    <p>Gholami, et. al. Number of Lymph Nodes Removed and Survival after
-Gastric Cancer Resection: An Analysis from the US Gastric Cancer Collaborative.
-J Am Coll Surg. 2015 Aug;221(2):291-9. doi: 10.1016/j.jamcollsurg.2015.04.024.</p>
-
-  </div>
-  <div class="col-md-3 hidden-xs hidden-sm" id="pkgdown-sidebar">
-    <nav id="toc" data-toggle="toc" class="sticky-top">
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-  <p>Developed by <a href='http://statweb.stanford.edu/~ckirby/brad/'>Bradley Efron</a>, <a href='https://web.stanford.edu/~naras'>Balasubramanian Narasimhan</a>.</p>
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-
-<div class="pkgdown">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/">pkgdown</a> 1.5.1.</p>
-</div>
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From 902b2e66088469637faac2c780ca7899b8b76906 Mon Sep 17 00:00:00 2001
From: Balasubramanian Narasimhan <b.naras@gmail.com>
Date: Tue, 4 Aug 2026 17:29:18 -0700
Subject: [PATCH 3/3] Add hex logo; update pkgdown config for Bootstrap 5 and
 MathJax

---
 _pkgdown.yml                                 |  12 ++-
 man/figures/logo.png                         | Bin 0 -> 37304 bytes
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 pkgdown/favicon/apple-touch-icon.png         | Bin 0 -> 7628 bytes
 pkgdown/favicon/favicon-96x96.png            | Bin 0 -> 3474 bytes
 pkgdown/favicon/favicon.ico                  | Bin 0 -> 15086 bytes
 pkgdown/favicon/favicon.svg                  |  12 +++
 pkgdown/favicon/site.webmanifest             |  21 +++++
 pkgdown/favicon/web-app-manifest-192x192.png | Bin 0 -> 8231 bytes
 pkgdown/favicon/web-app-manifest-512x512.png | Bin 0 -> 26745 bytes
 tools/logo/README.md                         |  26 ++++++
 tools/logo/gen_curves.py                     |  49 ++++++++++
 tools/logo/make_logos.py                     |  93 +++++++++++++++++++
 13 files changed, 229 insertions(+), 4 deletions(-)
 create mode 100644 man/figures/logo.png
 create mode 100644 man/figures/logo.svg
 create mode 100644 pkgdown/favicon/apple-touch-icon.png
 create mode 100644 pkgdown/favicon/favicon-96x96.png
 create mode 100644 pkgdown/favicon/favicon.ico
 create mode 100644 pkgdown/favicon/favicon.svg
 create mode 100644 pkgdown/favicon/site.webmanifest
 create mode 100644 pkgdown/favicon/web-app-manifest-192x192.png
 create mode 100644 pkgdown/favicon/web-app-manifest-512x512.png
 create mode 100644 tools/logo/README.md
 create mode 100644 tools/logo/gen_curves.py
 create mode 100644 tools/logo/make_logos.py

diff --git a/_pkgdown.yml b/_pkgdown.yml
index 872639f..7cae618 100644
--- a/_pkgdown.yml
+++ b/_pkgdown.yml
@@ -1,7 +1,11 @@
-authors:
+url: https://bnaras.github.io/deconvolveR/
 
-  Balasubramanian Narasimhan:
-    href: https://web.stanford.edu/~naras
+template:
+  bootstrap: 5
+  math-rendering: mathjax
 
+authors:
+  Balasubramanian Narasimhan:
+    href: https://naras.su.domains
   Bradley Efron:
-    href: http://statweb.stanford.edu/~ckirby/brad/
+    href: https://efron.ckirby.su.domains/
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diff --git a/man/figures/logo.svg b/man/figures/logo.svg
new file mode 100644
index 0000000..c56a4c6
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diff --git a/tools/logo/README.md b/tools/logo/README.md
new file mode 100644
index 0000000..4f9e17d
--- /dev/null
+++ b/tools/logo/README.md
@@ -0,0 +1,26 @@
+# Logo generation
+
+The deconvolveR hex logo (`man/figures/logo.svg` / `logo.png`) is generated
+by the two scripts here. The motif: the spiky prior g(θ) (solid indigo)
+recovered from the smooth observed density f (dotted), which is g convolved
+with normal noise.
+
+To regenerate:
+
+```sh
+python3 gen_curves.py > curves.txt   # density curves as SVG path data
+python3 make_logos.py                # writes logo-*.svg candidates here
+```
+
+`make_logos.py` produces four styled candidates; the package logo is the
+`minimal-light` variant (near-white ground, indigo curves, cardinal border).
+Install it with:
+
+```sh
+cp logo-minimal-light.svg ../../man/figures/logo.svg
+rsvg-convert -w 480 logo-minimal-light.svg -o ../../man/figures/logo.png
+Rscript -e 'pkgdown::build_favicons(overwrite = TRUE)'
+```
+
+This directory is listed in `.Rbuildignore` and is not shipped in the
+package tarball.
diff --git a/tools/logo/gen_curves.py b/tools/logo/gen_curves.py
new file mode 100644
index 0000000..8d32370
--- /dev/null
+++ b/tools/logo/gen_curves.py
@@ -0,0 +1,49 @@
+"""Generate SVG path data for deconvolveR logo curves.
+
+g(theta): spiky bimodal prior (mixture of two narrow normals).
+f(x):     g convolved with a wide normal kernel (the observed, blurred density).
+Both scaled into hex-sticker coordinates (viewBox 240x278, pointy-top hexagon).
+"""
+import math
+
+N = 121
+
+
+def norm(x, mu, sd):
+    return math.exp(-0.5 * ((x - mu) / sd) ** 2) / (sd * math.sqrt(2 * math.pi))
+
+
+def densities():
+    xs = [i / (N - 1) for i in range(N)]
+    g = [0.62 * norm(x, 0.33, 0.055) + 0.38 * norm(x, 0.72, 0.045) for x in xs]
+    # convolution of the mixture with N(0, 0.13) kernel is analytic:
+    f = [0.62 * norm(x, 0.33, math.hypot(0.055, 0.13))
+         + 0.38 * norm(x, 0.72, math.hypot(0.045, 0.13)) for x in xs]
+    return xs, g, f
+
+
+def to_path(xs, ys, x0, x1, ybase, ymax_px, ymax_val, close=False):
+    pts = []
+    for x, y in zip(xs, ys):
+        px = x0 + x * (x1 - x0)
+        py = ybase - (y / ymax_val) * ymax_px
+        pts.append((round(px, 1), round(py, 1)))
+    d = "M{},{}".format(*pts[0]) + " L" + " ".join(f"{px},{py}" for px, py in pts[1:])
+    if close:
+        d += f" L{pts[-1][0]},{ybase} L{pts[0][0]},{ybase} Z"
+    return d
+
+
+xs, g, f = densities()
+gmax = max(g)
+
+X0, X1 = 38, 202          # horizontal extent inside hex
+BASE = 186                # curve baseline y
+print("== g curve (line) ==")
+print(to_path(xs, g, X0, X1, BASE, 96, gmax))
+print("== g curve (closed area) ==")
+print(to_path(xs, g, X0, X1, BASE, 96, gmax, close=True))
+print("== f curve (line) ==")
+print(to_path(xs, f, X0, X1, BASE, 96, gmax))
+print("== f curve (closed area) ==")
+print(to_path(xs, f, X0, X1, BASE, 96, gmax, close=True))
diff --git a/tools/logo/make_logos.py b/tools/logo/make_logos.py
new file mode 100644
index 0000000..f1af4ce
--- /dev/null
+++ b/tools/logo/make_logos.py
@@ -0,0 +1,93 @@
+"""Assemble four deconvolveR hex-logo candidates from generated curve paths."""
+import re, pathlib
+
+d = pathlib.Path(__file__).parent
+txt = (d / "curves.txt").read_text()
+blocks = re.split(r"== .+? ==\n", txt)[1:]
+G_LINE, G_AREA, F_LINE, F_AREA = [b.strip().replace("\n", " ") for b in blocks]
+
+# pointy-top hexagon, viewBox 240x278, center (120,139)
+HEX = "M120,3 L237.8,71 L237.8,207 L120,275 L2.2,207 L2.2,71 Z"
+HEX_INNER = "M120,10 L231.7,74.5 L231.7,203.5 L120,268 L8.3,203.5 L8.3,74.5 Z"
+
+FONT = "'Avenir Next','Avenir','Futura','Century Gothic','Trebuchet MS',sans-serif"
+
+def svg(body, bg_defs=""):
+    return f'''<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 240 278" role="img" aria-label="deconvolveR hex logo">
+  <defs>
+    {bg_defs}
+    <clipPath id="hexclip"><path d="{HEX}"/></clipPath>
+  </defs>
+{body}
+</svg>'''
+
+def wordmark(y, fill, r_fill, size=25, weight=600):
+    return (f'<text x="120" y="{y}" text-anchor="middle" font-family="{FONT}" '
+            f'font-size="{size}" font-weight="{weight}" fill="{fill}" '
+            f'letter-spacing="0.5">deconvolve<tspan fill="{r_fill}" font-weight="800">R</tspan></text>')
+
+logos = {}
+
+# 1. Hidden Signal - deep navy, ghost f, teal g
+logos["hidden-signal"] = svg(f'''
+  <path d="{HEX}" fill="url(#navy)"/>
+  <g clip-path="url(#hexclip)">
+    <path d="{F_AREA}" fill="#7FB3E8" fill-opacity="0.14"/>
+    <path d="{F_LINE}" fill="none" stroke="#7FB3E8" stroke-opacity="0.55" stroke-width="2" stroke-dasharray="1 5" stroke-linecap="round"/>
+    <path d="{G_AREA}" fill="#2EE6C8" fill-opacity="0.13"/>
+    <path d="{G_LINE}" fill="none" stroke="#2EE6C8" stroke-width="3.5" stroke-linejoin="round" stroke-linecap="round"/>
+    <line x1="30" y1="186" x2="210" y2="186" stroke="#7FB3E8" stroke-opacity="0.35" stroke-width="1"/>
+  </g>
+  {wordmark(224, "#EAF4FF", "#2EE6C8")}
+  <text x="120" y="246" text-anchor="middle" font-family="{FONT}" font-size="11" font-weight="500" fill="#7FB3E8" letter-spacing="2">g-MODELING</text>
+  <path d="{HEX}" fill="none" stroke="#2EE6C8" stroke-width="3"/>
+''', '<linearGradient id="navy" x1="0" y1="0" x2="0" y2="1"><stop offset="0" stop-color="#123A63"/><stop offset="1" stop-color="#0A1F38"/></linearGradient>')
+
+# 2. Cardinal - Stanford red, cream curves
+logos["cardinal"] = svg(f'''
+  <path d="{HEX}" fill="#8C1515"/>
+  <g clip-path="url(#hexclip)">
+    <path d="{F_LINE}" fill="none" stroke="#F7F0E3" stroke-opacity="0.5" stroke-width="2" stroke-dasharray="1 5" stroke-linecap="round"/>
+    <path d="{G_AREA}" fill="#F7F0E3" fill-opacity="0.22"/>
+    <path d="{G_LINE}" fill="none" stroke="#F7F0E3" stroke-width="3.5" stroke-linejoin="round" stroke-linecap="round"/>
+    <line x1="30" y1="186" x2="210" y2="186" stroke="#F7F0E3" stroke-opacity="0.4" stroke-width="1"/>
+  </g>
+  {wordmark(226, "#F7F0E3", "#F7F0E3", weight=600)}
+  <path d="{HEX_INNER}" fill="none" stroke="#F7F0E3" stroke-opacity="0.55" stroke-width="1.2"/>
+  <path d="{HEX}" fill="none" stroke="#5E0E0E" stroke-width="3"/>
+''')
+
+# 3. Minimal Light - near-white, indigo fine lines
+logos["minimal-light"] = svg(f'''
+  <path d="{HEX}" fill="#FAFAF6"/>
+  <g clip-path="url(#hexclip)">
+    <g transform="translate(0,-16)">
+      <path d="{F_AREA}" fill="#8B87D8" fill-opacity="0.16"/>
+      <path d="{F_LINE}" fill="none" stroke="#8B87D8" stroke-width="1.8" stroke-dasharray="1 4.5" stroke-linecap="round"/>
+      <path d="{G_LINE}" fill="none" stroke="#37339B" stroke-width="3" stroke-linejoin="round" stroke-linecap="round"/>
+      <line x1="30" y1="186" x2="210" y2="186" stroke="#37339B" stroke-opacity="0.3" stroke-width="1"/>
+      <text x="203" y="200" font-family="Georgia,serif" font-style="italic" font-weight="bold" font-size="19" fill="#37339B" fill-opacity="0.85">&#952;</text>
+    </g>
+  </g>
+  {wordmark(214, "#2A2765", "#37339B", weight=500)}
+  <path d="{HEX}" fill="none" stroke="#8C1515" stroke-width="5"/>
+''')
+
+# 4. Amber Glow - charcoal, glowing amber g
+logos["amber-glow"] = svg(f'''
+  <path d="{HEX}" fill="#1C1B22"/>
+  <g clip-path="url(#hexclip)">
+    <path d="{F_LINE}" fill="none" stroke="#9A97A6" stroke-opacity="0.6" stroke-width="2" stroke-dasharray="1 5" stroke-linecap="round"/>
+    <path d="{G_LINE}" fill="none" stroke="#F5A623" stroke-width="6" stroke-opacity="0.28" filter="url(#soft)" stroke-linejoin="round"/>
+    <path d="{G_AREA}" fill="#F5A623" fill-opacity="0.10"/>
+    <path d="{G_LINE}" fill="none" stroke="#F5A623" stroke-width="3.2" stroke-linejoin="round" stroke-linecap="round"/>
+    <line x1="30" y1="186" x2="210" y2="186" stroke="#9A97A6" stroke-opacity="0.4" stroke-width="1"/>
+  </g>
+  {wordmark(224, "#F2EFE8", "#F5A623")}
+  <text x="120" y="246" text-anchor="middle" font-family="Georgia,serif" font-style="italic" font-size="13" fill="#9A97A6">f = g &#8727; noise</text>
+  <path d="{HEX}" fill="none" stroke="#F5A623" stroke-width="3"/>
+''', '<filter id="soft" x="-20%" y="-20%" width="140%" height="140%"><feGaussianBlur stdDeviation="3"/></filter>')
+
+for name, content in logos.items():
+    (d / f"logo-{name}.svg").write_text(content)
+    print(f"logo-{name}.svg  {len(content)} bytes")