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Merge pull request #263 from avehtari/lecture_11a
Lecture 11a
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‎slides/BDA_lecture_11a.pdf‎

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‎slides/BDA_lecture_11a.tex‎

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\documentclass[t]{beamer}
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%\documentclass[finnish,english,handout]{beamer}
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\usepackage{tikz}
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\usepackage{bm}
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\usepackage[T1]{fontenc}
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\usepackage{booktabs}
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\usepackage[utf8]{inputenc}
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\usepackage{newtxtext} % times
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%\usepackage[scaled=.95]{cabin} % sans serif
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\usepackage{amsmath}
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\usepackage[varqu,varl]{inconsolata} % typewriter
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\usepackage[varg]{newtxmath}
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\usefonttheme[onlymath]{serif} % beamer font theme
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\usepackage{microtype}
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\usepackage{afterpage}
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\usepackage{url}
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\urlstyle{same}
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% \usepackage{amsbsy}
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% \usepackage{eucal}
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\usepackage{rotating}
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\usepackage{listings}
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\usepackage{lstbayes}
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\usepackage[all,poly,ps,color]{xy}
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\usepackage{eurosym}
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\usepackage{natbib}
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\bibliographystyle{apalike}
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\mode<presentation>
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{
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\setbeamercovered{invisible}
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\setbeamertemplate{itemize items}[circle]
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\setbeamercolor{frametitle}{bg=white,fg=navyblue}
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\setbeamertemplate{navigation symbols}{}
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\setbeamertemplate{headline}[default]{}
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\setbeamertemplate{footline}[split]
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% \setbeamertemplate{headline}[text line]{\insertsection}
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\setbeamertemplate{footline}[frame number]
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}
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\pdfinfo{
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/Title (BDA, Lecture 11a)
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/Author (Aki Vehtari) %
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/Keywords (Bayesian data analysis)
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}
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\definecolor{forestgreen}{rgb}{0.1333,0.5451,0.1333}
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\definecolor{navyblue}{rgb}{0,0,0.5}
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\definecolor{list1}{rgb}{0,0.2549,0.6784}
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\renewcommand{\emph}[1]{\textcolor{navyblue}{#1}}
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\definecolor{set11}{HTML}{E41A1C}
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\definecolor{set12}{HTML}{377EB8}
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\definecolor{set13}{HTML}{4DAF4A}
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\definecolor{prior}{RGB}{102,194,165}
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\definecolor{likelihood}{RGB}{252,141,98}
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\graphicspath{{./figs/}}
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\parindent=0pt
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\parskip=8pt
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\tolerance=9000
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\abovedisplayshortskip=0pt
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%\renewcommand{\itemsep}{0pt}
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% Lists
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\newenvironment{list1}{
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\begin{list}{$\color{list1}\bullet$}{\itemsep=6pt}}{
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\end{list}}
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\newenvironment{list1s}{
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\begin{list}{$\includegraphics[width=5pt]{logo.eps}$}{\itemsep=6pt}}{
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\end{list}}
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\newenvironment{list2}{
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\begin{list}{-}{\baselineskip=12pt\itemsep=2pt}}{
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\end{list}}
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\newenvironment{list3}{
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\begin{list}{$\cdot$}{\baselineskip=15pt}}{
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\end{list}}
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\def\o{{\mathbf o}}
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\def\t{{\mathbf \theta}}
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\def\w{{\mathbf w}}
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\def\x{{\mathbf x}}
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\def\y{{\mathbf y}}
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\def\z{{\mathbf z}}
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\def\peff{p_{\mathrm{eff}}}
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\def\eff{\mathrm{eff}}
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\DeclareMathOperator{\E}{E}
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\DeclareMathOperator{\Var}{Var}
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\DeclareMathOperator{\var}{var}
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\DeclareMathOperator{\Sd}{Sd}
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\DeclareMathOperator{\sd}{sd}
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\DeclareMathOperator{\Gammad}{Gamma}
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\DeclareMathOperator{\Invgamma}{Inv-gamma}
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\DeclareMathOperator{\Negbin}{Neg-bin}
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\DeclareMathOperator{\normal}{normal}
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\DeclareMathOperator{\gammadist}{gamma}
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\DeclareMathOperator{\expdist}{exponential}
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\DeclareMathOperator{\Poisson}{Poisson}
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\DeclareMathOperator{\betadist}{Beta}
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\DeclareMathOperator{\logit}{logit}
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\DeclareMathOperator{\N}{N}
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\DeclareMathOperator{\U}{U}
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\DeclareMathOperator{\BF}{BF}
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\DeclareMathOperator{\Invchi2}{Inv-\chi^2}
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\DeclareMathOperator{\NInvchi2}{N-Inv-\chi^2}
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\DeclareMathOperator{\InvWishart}{Inv-Wishart}
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\DeclareMathOperator{\tr}{tr}
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% \DeclareMathOperator{\Pr}{Pr}
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\def\euro{{\footnotesize \EUR\, }}
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\DeclareMathOperator{\rep}{\mathrm{rep}}
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% \def\dashxy(#1){%
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% /xydash{[#1] 0 setdash}def}
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% \def\grayxy(#1){%
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% /xycolor{#1 setgray}def}
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% \newgraphescape{D}[1]{!{\ar @*{[!\dashxy(2 2)]} "#1"}}
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% \newgraphescape{P}[1]{!{\ar "#1"}}
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% \newgraphescape{F}[1]{!{*+=<2em>[F=]{#1}="#1"}}
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% \newgraphescape{O}[1]{!{*+=<2em>[F]{#1}="#1"}}
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% \newgraphescape{V}[1]{!{*+=<2em>[o][F]{#1}="#1"}}
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% \newgraphescape{B}[3]{!{{ "#1"*+#3\frm{} }.{ "#2"*+#3\frm{} } *+[F:!\grayxy(0.75)]\frm{}}}
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\title[]{Bayesian data analysis}
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\subtitle{}
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\author{Noa Kallioinen}
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\institute[Aalto]{}
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\date[]{}
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%\beamerdefaultoverlayspecification{<+->}
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\begin{document}
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%\maketitle
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\begin{frame}{Prior and likelihood sensitivity checks with priorsense}
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\begin{itemize}[<+->]
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\item sensible prior specification is important, as otherwise priors can strongly and unintentionally influence results
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\item we recommend checking prior and likelihood sensitivity using priorsense (Kallioinen et al., 2023)
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\end{itemize}
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\begin{minipage}{0.2\textwidth}
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\onslide<2->{\begin{figure}
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\includegraphics[width=\textwidth]{logo.png}
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\end{figure}}
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\end{minipage}
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\hspace{0.05\textwidth}
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\begin{minipage}{0.65\textwidth}
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\begin{itemize}[<+->]
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\item efficient prior and likelihood sensitivity checks
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\item provides numerical and graphical diagnostics
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\item compatible with \texttt{brms} and applicable to any model (excluding flat priors)
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\item no need to refit
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\end{itemize}
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\end{minipage}
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\end{frame}
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\begin{frame}{How it works}
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\begin{minipage}{0.2\textwidth}
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\begin{figure}
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\includegraphics[width=\linewidth]{logo.png}
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\end{figure}
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\end{minipage}
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\hspace{0.05\textwidth}
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\begin{minipage}{0.65\textwidth}
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\begin{itemize}[<+->]
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\item modifies the posterior by power-scaling the \textcolor{prior}{\textbf{prior}} or \textcolor{likelihood}{\textbf{likelihood}} slightly
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\item original posterior: \(p(\theta\mid y) \propto p(\theta)p(y\mid \theta)\)
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\item modified posterior: \(\bm{{\color{prior}{p(\theta)^{\alpha}}}} p(y\mid \theta)\) or \(p(\theta)
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\bm{{\color{likelihood}{p(y\mid \theta)^{\alpha}}}}\)
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\item uses Pareto-smoothed importance sampling to estimate modified posterior from original posterior draws \(\theta^{(s)}\)
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\end{itemize}
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\end{minipage}
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\end{frame}
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\begin{frame}{Power-scaling}
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\begin{figure}
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\centering
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\input{tikz/example_dists}
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\caption{Example power-scaled distributions}
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\label{fig:placeholder}
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\end{figure}
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\end{frame}
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\begin{frame}{Interpretation}
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\begin{figure}
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\centering
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\input{tikz/conflict-example}
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\caption{Prior-likelihood conflict}
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\label{fig:placeholder}
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\end{figure}
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\end{frame}
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\begin{frame}{Computation}
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\begin{itemize}
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\onslide<1->{\item[] For prior power-scaling:
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\item[] \(w_{pri}^{(s)} = \dfrac{\bm{{\color{prior}{p(\theta^{(s)})^{\alpha}}}}p(\theta^{(s)} \mid y)}{p(\theta^{(s)})p(\theta^{(s)} \mid y)} = \dfrac{\bm{{\color{prior}{p(\theta^{(s)})^{\alpha}}}}}{p(\theta^{(s)})} = p(\theta^{(s)})^{(\alpha - 1)}\)
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\item[]
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\item[]
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}
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\onslide<2>{\item[] For likelihood power-scaling:
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\item[] \(w_{lik}^{(s)} = \dfrac{p(\theta^{(s)})\bm{{\color{likelihood}{p(\theta^{(s)} \mid y)^{\alpha}}}}}{p(\theta^{(s)})p(\theta^{(s)} \mid y)} = \dfrac{\bm{{\color{likelihood}{p(\theta^{(s)} \mid y)^{\alpha}}}}}{p(y \mid \theta^{(s)})} = p(y \mid \theta^{(s)})^{(\alpha - 1)}\)}
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\end{itemize}
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\end{frame}
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\begin{frame}{Example}
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\begin{itemize}
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\item Dataset: \texttt{npk}
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\item Goal: Model effect of Nitrogen, Phosphate and Potassium on pea crop yield
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\item Data: 24 observations of crop yield (3 observations of each combination)
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\end{itemize}
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\begin{figure}
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\centering
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\includegraphics[width=0.75\linewidth]{npk.png}
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\caption{Mean yield per combination}
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\label{fig:placeholder}
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\end{figure}
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\end{frame}
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\begin{frame}{Example}
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Model:
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\begin{align*}
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\onslide<1->{y_i &\sim \N(\mu_i, \sigma^2)} \\[0.5em]
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\onslide<2->{\mu_i =& \beta_{\text{Intercept}}} \\
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\onslide<3->{ \quad +& \beta_{\text{N}} N_i && \text{(main effects)} \\
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\quad +& \beta_{\text{P}} P_i \\
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\quad +& \beta_{\text{K}} K_i
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} \\
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\onslide<4->{ \quad +& \beta_{\text{N} \times \text{K}} N_i K_i
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&& \text{(2-way interactions)}} \\
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\onslide<4->{ \quad +& \beta_{\text{N} \times \text{P}} N_i P_i} \\
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\onslide<4->{ \quad +& \beta_{\text{K} \times \text{P}} K_i P_i} \\
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% \onslide<5->{ &\quad + \beta_{\text{N} \times \text{P} \times \text{K}}
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% N_i P_i K_i
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% && \text{(3-way interaction)}} \\
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\end{align*}
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\end{frame}
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\begin{frame}{Example: Initial model}
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\onslide<1->{
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Main effects only
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\texttt{brms} formula:
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\texttt{yield} \sim \texttt{N + P + K }
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}
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\onslide<2->{
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\texttt{prior(normal(0, 2.5), coef = "b\_N1", \textcolor{blue}{tag = "main"})}
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\texttt{prior(normal(0, 2.5), coef = "b\_K1", \textcolor{blue}{tag = "main"})}
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\texttt{prior(normal(0, 2.5), coef = "b\_P1", \textcolor{blue}{tag = "main"})}
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\textcolor{blue}{tagging} a group of priors allows for selective power-scaling
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}
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\end{frame}
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\begin{frame}{Example: Initial model checks}
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\only<1-2>{
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\texttt{powerscale\_plot\_dens(fit, \textcolor{blue}{prior\_selection = "main"})}
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}
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\only<2>{
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{npk_model1.png}
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\end{figure}
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}
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\only<3-4>{
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\texttt{powerscale\_sensitivity(fit, \textcolor{blue}{prior\_selection = "main"}})
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}
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\only<4>{
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\begin{table}[]
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\centering
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\begin{tabular}{llll}
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variable & prior sens. & likelihood sens. & diagnosis \\
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\midrule
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b\_N1 & 0.15 & 0.18 & potential prior-data conflict \\
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b\_P1 & 0.06 & 0.07 & potential prior-data conflict \\
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b\_K1 & 0.11 & 0.14 & potential prior-data conflict
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\end{tabular}
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\caption{Sensitivity diagnostics}
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\label{tab:placeholder}
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\end{table}
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}
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\end{frame}
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\begin{frame}{Example: Adjusted prior}
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\texttt{prior(normal(0, \textcolor{blue}{25}), coef = "b\_N1", tag = "main")}
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\texttt{prior(normal(0, \textcolor{blue}{25}), coef = "b\_K1", tag = "main")}
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\texttt{prior(normal(0, \textcolor{blue}{25}), coef = "b\_P1", tag = "main")}
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\end{frame}
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\begin{frame}{Example: Adjusted prior checks}
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\only<1>{
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\begin{figure}
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\centering
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\includegraphics[width=1\linewidth]{npk_model1adj.png}
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\end{figure}
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}
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\only<2>{
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\begin{table}[]
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\centering
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\begin{tabular}{llll}
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variable & prior sens. & likelihood sens. & diagnosis \\
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\midrule
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b\_N1 & 0.00 & 0.10 & - \\
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b\_P1 & 0.00 & 0.09 & - \\
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b\_K1 & 0.00 & 0.10 & -
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\end{tabular}
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\end{table}
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}
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\end{frame}
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\begin{frame}{Example: Second model}
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\onslide<1->{
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Main effects and two-way interactions
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\texttt{brms} formula:
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\texttt{yield} \sim \texttt{(N + P + K)\^{}2}
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}
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\onslide<2->{
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\texttt{prior(normal(0, 25), coef = "b$\_$N1:P1", \textcolor{blue}{tag = "twoway"})}
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\texttt{prior(normal(0, 25), coef = "b$\_$N1:K1", \textcolor{blue}{tag = "twoway"})}
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\texttt{prior(normal(0, 25), coef = "b$\_$P1:K1", \textcolor{blue}{tag = "twoway"})}
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}
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\end{frame}
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\begin{frame}{Example: Second model checks}
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\begin{figure}
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\centering
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\includegraphics[width=1\linewidth]{npk_model2.png}
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\end{figure}
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\end{frame}
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\begin{frame}{Summary}
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\begin{itemize}[<+->]
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\item priors can be challenging to specify without domain knowledge
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\item prior and likelihood sensitivity checking is recommended
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\item \texttt{priorsense} can make sensitivity checking faster
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\item with many parameters, use tags to select priors and look at predictions or model fit measures
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\item if checks indicate strong prior sensitivity, prior-data conflict, or weak likelihood, further thought should be put into the prior
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\item see more at \url{n-kall.github.io/priorsense}
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\end{itemize}
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\end{frame}
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\end{document}

‎slides/figs/logo.png‎

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‎slides/figs/npk.png‎

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‎slides/figs/npk_model1.png‎

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‎slides/figs/npk_model1adj.png‎

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‎slides/figs/npk_model2.png‎

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