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00000000..9a28571d --- /dev/null +++ b/chapter_notes/BDA_notes_ch13.tex @@ -0,0 +1,138 @@ +\documentclass[a4paper,11pt,english]{article} + +\usepackage{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{amsmath} +\usepackage{microtype} +\usepackage{url} +\urlstyle{same} + +\usepackage[bookmarks=false]{hyperref} +\hypersetup{% + bookmarksopen=true, + bookmarksnumbered=true, + pdftitle={Bayesian data analysis}, + pdfsubject={Comments}, + pdfauthor={Aki Vehtari}, + pdfkeywords={Bayesian probability theory, Bayesian inference, Bayesian data analysis}, + pdfstartview={FitH -32768} +} + + +% if not draft, smaller printable area makes the paper more readable +\topmargin -4mm +\oddsidemargin 0mm +\textheight 225mm +\textwidth 160mm + +%\parskip=\baselineskip +\def\eff{\mathrm{rep}} + +\DeclareMathOperator{\E}{E} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\var}{var} +\DeclareMathOperator{\Sd}{Sd} +\DeclareMathOperator{\sd}{sd} +\DeclareMathOperator{\Bin}{Bin} +\DeclareMathOperator{\Beta}{Beta} +\DeclareMathOperator{\Invchi2}{Inv-\chi^2} +\DeclareMathOperator{\NInvchi2}{N-Inv-\chi^2} +\DeclareMathOperator{\logit}{logit} +\DeclareMathOperator{\N}{N} +\DeclareMathOperator{\U}{U} +\DeclareMathOperator{\tr}{tr} +%\DeclareMathOperator{\Pr}{Pr} +\DeclareMathOperator{\trace}{trace} +\DeclareMathOperator{\rep}{\mathrm{rep}} + +\pagestyle{empty} + +\begin{document} +\thispagestyle{empty} + +\section*{Bayesian data analysis -- reading instructions 13} +\smallskip +{\bf Aki Vehtari} +\smallskip + +\subsection*{Chapter 13: Modal and distributional approximations} + +Chapter 4 presented normal distribution approximation at the mode (aka +Laplace approximation). Chapter 13 discusses more about distributional +approximations. + +Outline of the chapter 13 +\begin{list}{$\bullet$}{\parsep=0pt\itemsep=2pt} +\item[13.1] Finding posterior modes + \begin{itemize} + \item[-] Newton's method is very fast if the distribution is close to + normal and the computation of the second derivatives is fast + \item[-] Stan uses limited-memory Broyden-Fletcher-Goldfarb-Shannon + (L-BFGS) which is a quasi-Newton method which needs only the first + derivatives (provided by Stan autodiff). L-BFGS is known for good + performance for wide variety of functions. + \end{itemize} +\item[13.2] Boundary-avoiding priors for modal summaries + \begin{itemize} + \item[-] Although full integration is preferred, sometimes optimization + of some parameters may be sufficient and faster, and then + boundary-avoiding priors maybe useful. + \end{itemize} +\item[13.3] Normal and related mixture approximations + \begin{itemize} + \item[-] Discusses how the normal approximation can be used to + approximate integrals of a a smooth function times the posterior. + \item[-] Discusses mixture and $t$ approximations. + \end{itemize} +\item[13.4] Finding marginal posterior modes using EM + \begin{itemize} + \item[-] Expectation maximization is less important in the time of + efficient probabilistic programming frameworks, but can be + sometimes useful for extra efficiency. + \end{itemize} +\item[13.5] Conditional and marginal posterior approximations + \begin{itemize} + \item[-] Even in the time of efficient probabilistic programming, the + methods discussed in this section can produce very big speedups + for a big set of commonly used models. The methods discussed are + important part of popular INLA software and are coming also to + Stan to speedup latent Gaussian variable models. + \end{itemize} +\item[13.6] Example: hierarchical normal model +\item[13.7] Variational inference + \begin{itemize} + \item[-] Variational inference (VI) is very popular in machine + learning, and this section presents it in terms of BDA. Auto-diff + variational inference in Stan was developed after BDA3 was + published. + \end{itemize} +\item[13.8] Expectation propagation + \begin{itemize} + \item[-] Practical efficient computation for expectation propagation + (EP) is applicable for more limited set of models than post-BDA3 + black-box VI, but for those models EP provides better posterior + approximation. Variants of EP can be used for parallelization of + any Bayesian computation for hierarchical models. + \end{itemize} +\item[13.9] Other approximations + \begin{itemize} + \item[-] Just brief mentions of INLA (uses methods discussed in 13.5), + CCD (deterministic adaptive quadrature approach) and ABC + (inference when you can only sample from the generative model). + \end{itemize} +\item[13.10] Unknown normalizing factors + \begin{itemize} + \item[-] Often the normalizing factor is not needed, but it can be + estimated using importance, bridge or path sampling. + \end{itemize} +\end{list} + + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/chapter_notes/BDA_notes_ch8.pdf b/chapter_notes/BDA_notes_ch8.pdf new file mode 100644 index 0000000000000000000000000000000000000000..91db54a5db64afb9716f09a1c65edb9ea4aa95a5 GIT binary patch literal 36924 zcma&MQ;;r7&}iAVZQHhX+uCj0wryj#ZQHhO+cv+(oWUP45%)&S+U)(_sVQNTC^dTvqi_6_7R545 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