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pdfkeywords={Bayesian probability theory, Bayesian inference, Bayesian data analysis}, + pdfstartview={FitH -32768}, + colorlinks=true, + linkcolor=navyblue, + citecolor=black, + filecolor=black, + urlcolor=blue +} + + +% 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 Part IV} +\smallskip +{\bf Aki Vehtari} +\smallskip +\bigskip + +\noindent +Part IV, Chapters 14--18 discuss basics of linear and generalized +linear models with several examples. The parts discussing computation +can be useful to provide additional insight on these models or +sometimes for actual computation, it's likely that most of the readers +will use some probabilistic programming framework for +computation. Regression and other stories (ROS) by Gelman, Hill and +Vehtari discusses linear and generalized linear models from the +modeling perspective more thoroughly. + +\subsection*{Chapter 14: Introduction to regression models} + +Outline of the chapter 14: +\begin{list}{$\bullet$}{\parsep=0pt\itemsep=2pt} +\item[14.1] Conditional modeling + \begin{itemize} + \item formal justification of conditional modeling + \item if joint model factorizes $p(y,x|\theta,\phi)={\color{blue}p(y|x,\theta)}p(x|\phi)$\\ + we can model just ${\color{blue}p(y|x,\theta)}$ + \end{itemize} +\item[14.2] Bayesian analysis of classical regression + \begin{itemize} + \item uninformative prior on $\beta$ and $\sigma^2$ + \item connection to multivariate normal (cf. Chapter 3) is useful to understand as it then reveals what would be the conjugate prior + \item closed form posterior and posterior predictive distribution + \item these properties are sometimes useful and thus good to know, + but with probabilistic programming less often needed + \end{itemize} +\item[14.3] Regression for causal inference: incumbency and voting + \begin{itemize} + \item Modelling example with bit of discussion on causal inference + (see more in ROS Chs. 18-21) + \end{itemize} +\item[14.4] Goals of regression analysis + \begin{itemize} + \item discussion of what we can do with regression analysis (see + more in ROS) + \end{itemize} +\item[14.5] Assembling the matrix of explanatory variables + \begin{itemize} + \item transformations, nonlinear relations, indicator variables, + interactions (see more in ROS) + \end{itemize} +\item[14.6] Regularization and dimension reduction + \begin{itemize} + \item a bit outdated and short (Bayesian Lasso is not a good idea), + see more in lecture 9.3, + \url{https://avehtari.github.io/modelselection/} and + \url{https://betanalpha.github.io/assets/case_studies/bayes_sparse_regression.html}) + \end{itemize} +\item[14.7] Unequal variances and correlations + \begin{itemize} + \item useful concept, but computation is easier with probabilistic + programming frameworks + \end{itemize} +\item[14.8] Including numerical prior information + \begin{itemize} + \item useful conceptually, but easy computation with probabilistic + programming frameworks makes it easier to define prior information + as the prior doesn't need to be conjugate + \item see more about priors in \url{https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations} + \end{itemize} +\end{list} + +\subsection*{Chapter 15 Hierarchical linear models} + +Chapter 15 combines hierarchical models from Chapter 5 and linear +models from Chapter 14. The chapter discusses some computational +issues, but probabilistic programming frameworks make computation for +hierarchical linear models easy. + +\vspace{\baselineskip} +\noindent +Outline of the chapter 15: +\begin{list}{$\bullet$}{\parsep=0pt\itemsep=2pt} +\item[15.1] Regression coefficients exchangeable in batches + \begin{itemize} + \item exchangeability of parameters + \item the discussion of fixed-, random- and mixed-effects models + is incomplete + \begin{itemize} + \item we don't recommend using these terms, but they are so + popular that it's useful to know them + \item a relevant comment is \emph{The terms ‘fixed’ and ‘random’ + come from the non-Bayesian statistical tradition and are + somewhat confusing in a Bayesian context where all unknown + parameters are treated as ‘random’ or, equivalently, as + having fixed but unknown values.} + \item often fixed effects correspond to population level + coefficients, random effects correspond to group or individual + level coefficients, and mixed model has both\\ + \begin{tabular}[t]{ll} + {\tt y $\sim$ 1 + x} & fixed / population effect; pooled model\\ + {\tt y $\sim$ 1 + (0 + x | g) } & random / group effects \\ + {\tt y $\sim$ 1 + x + (1 + x | g) } & mixed effects; hierarchical model + \end{tabular} + \end{itemize} + \end{itemize} + \item[15.2] Example: forecasting U.S. presidential elections + \begin{itemize} + \item illustrative example + \end{itemize} + \item[15.3] Interpreting a normal prior distribution as extra data + \begin{itemize} + \item includes very useful interpretation of hierarchical linear + model as a single linear model with certain design matrix + \end{itemize} + \item[15.4] Varying intercepts and slopes + \begin{itemize} + \item extends from hierarchical model for scalar parameter to + joint hierarchical model for several parameters + \end{itemize} + \item[15.5] Computation: batching and transformation + \begin{itemize} + \item Gibbs sampling part is mostly outdated + \item transformations for HMC is useful if you write your own + models, but the section is quite short and you can get more + information from Stan user guide 21.7 Reparameterization and + \url{https://mc-stan.org/users/documentation/case-studies/divergences_and_bias.html} + \end{itemize} + \item[15.6] Analysis of variance and the batching of coefficients + \begin{itemize} + \item ANOVA as Bayesian hierarchical linear model + \item rstanarm and brms packages make it easy to make ANOVA + \end{itemize} + \item[15.7] Hierarchical models for batches of variance components + \begin{itemize} + \item more variance components + \end{itemize} +\end{list} + +\subsection*{Chapter 16 Generalized linear models} + +Chapter 16 extends linear models to have non-normal observation +models. Model in Bioassay example in Chapter 3 is also generalized +linear model. Chapter reviews the basics and discusses some +computational issues, but probabilistic programming frameworks make +computation for generalized linear models easy (especially with +rstanarm and brms). Regression and other stories (ROS) by Gelman, Hill +and Vehtari discusses generalized linear models from the modeling +perspective more thoroughly. + + +\vspace{\baselineskip} +\noindent +Outline of the chapter 16: +\begin{list}{$\bullet$}{\parsep=0pt\itemsep=2pt} +\item[16 Intro:] + Parts of generalized linear model (GLM): + \begin{itemize} + \item[1.] The linear predictor $\eta = X\beta$ + \item[2.] The link function $g(\cdot)$ and $\mu = g^{-1}(\eta)$ + \item[3.] Outcome distribution model with location parameter $\mu$ + \begin{itemize} + \item the distribution can also depend on dispersion + parameter $\phi$ + \item originally just exponential family distributions + (e.g. Poisson, binomial, negative-binomial), which all have + natural location-dispersion parameterization + \item after MCMC made computation easy, GLM can refer to + models where outcome distribution is not part of exponential + family and dispersion parameter may have its own latent linear + predictor + \end{itemize} + \end{itemize} +\item[16.1] Standard generalized linear model likelihoods + \begin{itemize} + \item section title says ``likelihoods'', but it would be better to say ``observation models'' + \item continuous data: normal, gamma, Weibull mentioned, but common + are also Student's $t$, log-normal, log-logistic, and various + extreme value distributions like generalized Pareto distribution + \item binomial (Bernoulli as a special case) for binary and count + data with upper limit + \begin{itemize} + \item Bioassay model uses binomial observation model + \end{itemize} + \item Poisson for count data with no upper limit + \begin{itemize} + \item Poisson is useful approximation of Binomial when the observed + counts are much smaller than the upper limit + \end{itemize} + \end{itemize} + \item[16.2] Working with generalized linear models + \begin{itemize} + \item bit of this and that information on how think about GLMs (see + ROS for more) + \item normal approximation to the likelihood is good for thinking + how much information non-normal observations provide, can be + useful for someone thinking about computation, but easy + computation with probabilistic programming frameworks means not + everyone needs this + \end{itemize} + \item[16.3] Weakly informative priors for logistic regression + \begin{itemize} + \item an excellent section although the recommendation on using + Cauchy has changed (see + \url{https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations}) + \item the problem of separation is useful to understand + \item computation part is outdated as probabilistic programming + frameworks make the computation easy + \end{itemize} + \item[16.4] Overdispersed Poisson regression for police stops + \begin{itemize} + \item an example + \end{itemize} + \item[16.5] State-level opinions from national polls + \begin{itemize} + \item another example + \end{itemize} + \item[16.6] Models for multivariate and multinomial responses + \begin{itemize} + \item extension to multivariate responses + \item polychotomous data with multivariate binomial or Poisson + \item models for ordered categories + \end{itemize} + \item[16.7] Loglinear models for multivariate discrete data + \begin{itemize} + \item multinomial or Poisson as loglinear models + \end{itemize} + \end{list} + +\subsection*{Chapter 17 Models for robust inference} + +Chapter 17 discusses over-dispersed observation models. The discussion +is useful beyond generalized linear models. The computation is +outdated. See Regression and other stories (ROS) by Gelman, Hill +and Vehtari for more examples. + +\vspace{\baselineskip} +\noindent +Outline of the chapter 17: +\begin{list}{$\bullet$}{\parsep=0pt\itemsep=2pt} + \item[17.1] Aspects of robustness + \begin{itemize} + \item overdispersed models are often connected to robustness of + inferences to outliers, but the observed data can be overdispersed + without any observation being outlier + \item outlier is sensible only in the context of the model, being + something not well modelled or something requiring extra model + component + \item switching to generic overdispersed model can help to recognize + problem in the non-robust model (sensitivity analysis), but it + can also throw away useful information in the ``outliers'' and it + would be useful to think what is the generative mechanism for + observations which are not like others + \end{itemize} + \item[17.2] Overdispersed versions of standard models\\ + \begin{tabular}[t]{lcl}\small + normal & $\rightarrow$ & $t$-distribution\\ + Poisson & $\rightarrow$ & negative-binomial \\ + binomial & $\rightarrow$ & beta-binomial \\ + probit & $\rightarrow$ & logistic / robit + \end{tabular} + \item[17.3] Posterior inference and computation + \begin{itemize} + \item computation part is outdated as probabilistic programming + frameworks and MCMC make the computation easy + \item posterior is more likely to be multimodal + \end{itemize} + \item[17.4] Robust inference for the eight schools + \begin{itemize} + \item eight schools example is too small too see much difference + \end{itemize} + \item[17.5] Robust regression using t-distributed errors + \begin{itemize} + \item computation part is outdated as probabilistic programming + frameworks and MCMC make the computation easy + \item posterior is more likely to be multimodal + \end{itemize} +\end{list} + +\subsection*{Chapter 18 Models for missing data} + +Chapter 18 extends the data collection modelling from Chapter 8. See +Regression and other stories (ROS) by Gelman, Hill and Vehtari for +more examples. + +\vspace{\baselineskip} +\noindent +Outline of the chapter 18: +\begin{list}{$\bullet$}{\parsep=0pt\itemsep=2pt} + \item[18.1] Notation + \begin{itemize} + \item Missing completely at random (MCAR)\\ + missingness does not depend on missing values or other observed + values (including covariates) + \item Missing at random (MAR)\\ + missingness does not depend on missing values but may depend on + other observed values (including covariates) + \item Missing not at random (MNAR)\\ + missingness depends on missing values + \end{itemize} + \item[18.2] Multiple imputation + \begin{itemize} + \item[1.] make a model predicting missing data + \item[2.] sample repeatedly from the missing data model to generate + multiple imputed data sets + \item[3.] make usual inference for each imputed data set + \item[4.] combine results + \item discussion of computation is partially outdated + \end{itemize} + \item[18.3] Missing data in the multivariate normal and $t$ models + \begin{itemize} + \item a special continuous data case computation, which can still + be useful as fast starting point + \end{itemize} + \item[18.4] Example: multiple imputation for a series of polls + \begin{itemize} + \item an example + \end{itemize} + \item[18.5] Missing values with counted data + \begin{itemize} + \item discussion of computation for count data (ie computation in + 18.3 is not applicable) + \end{itemize} + \item[18.6] Example: an opinion poll in Slovenia + \begin{itemize} + \item another example + \end{itemize} +\end{list} + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: