slides.tex

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\usepackage{listings}
\usepackage{lstbayes} % Stan
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% if loaded after begin{document} a warning will appear: "pdfauthor already used"
\title[Estatística Bayesiana]{Estatística Bayesiana}
\author{Jose Storopoli \newline
        \texttt{josees@uni9.pro.br}}
\institute{Universidade Nove de Julho - UNINOVE}
\date{Maio 2022}

%%%% Maths crap
\newtheorem{theo}{Theorema}[]
\newtheorem{defn}{Definição}[]
\newtheorem{question}{Questão}[]
\newtheorem{idea}{Ideia}[]
\newtheorem{exemplo}{Exemplo}[]
\newtheorem{property}{Propriedade}[]

\begin{document}
%--- Title Page -------------------------------------------------------%

\maketitle

%--- Intro Bayes for Everyone ----------------------------------------%
\begingroup
  \AtBeginSection[]{}
  \include{intro.tex}
\endgroup

%--- Table of Contents-------------------------------------------------%
\setcounter{tocdepth}{1}
\begin{frame}[plain, noframenumbering]{Sumário}
  \tableofcontents
\end{frame}

%--- Distributions ----------------------------------------------------%

\tikzset{
    declare function={
      discreteuniform(\a,\b)=1/(\b-\a+1);%
      binomial(\n,\p)=\n!/(x!*(\n-x)!)*\p^x*(1-\p)^(\n-x);%
      poisson(\l)=(\l^x)*exp(-\l)/(x!);%
      negativebinomial(\r,\p)=((x+\r-1)!/((\r-1)!*x!))*((1-\p)^x*\p^\r);%
      continuousuniform(\a,\b)=1/(\b-\a);%
      gaussian(\m,\s)=1/(\s*sqrt(2*pi))*exp(-((x-\m)^2)/(2*\s^2));%
      lognormal(\m,\s)=1/(x*\s*sqrt(2*pi))*exp(-((ln(x)-\m)^2)/(2*\s^2));%
      exponential(\l)=\l*exp(-\l*x);%
      gamma(\z)=2.506628274631*sqrt(1/\z)+ 0.20888568*(1/\z)^(1.5)+ 0.00870357*(1/\z)^(2.5)- (174.2106599*(1/\z)^(3.5))/25920- (715.6423511*(1/\z)^(4.5))/1244160)*exp((-ln(1/\z)-1)*\z;%
      student(\n)=gamma((\n+1)/2.)/(sqrt(\n*pi)*gamma(\n/2.))*((1+(x*x)/\n)^(-(\n+1)/2.));%
      beta(\a,\b)=(x^(\a-1)*(1-x)^(\b-1)/((gamma(\a)*gamma(\b))/gamma(\a+\b));%
      binormal(\ma,\sa,\mb,\sb,\ro)=exp(-(((x-\ma)/\sa)^2+((y-\mb)/\sb)^2-(2*\ro)*((x-\ma)/\sa)*((y-\mb)/\sb))/(2*(1-\ro^2)))/(2*pi*\sa*\sb*(1-\ro^2)^0.5);%
      conditionalbinormal(\yc,\ma,\sa,\mb,\sb,\ro)=exp(-(((x-\ma)/\sa)^2+((\yc-\mb)/\sb)^2-(2*\ro)*((x-\ma)/\sa)*((\yc-\mb)/\sb))/(2*(1-\ro^2)))/(2*pi*\sa*\sb*(1-\ro^2)^0.5);%
      sumtwonormals(\ma,\sa,\wa,\mb,\sb,\wb)=(\wa*gaussian(\ma,\sa))+(\wb*gaussian(\mb,\sb));%
      normcdf(\m,\s)=1/(1 + exp(-0.07056*((x-\m)/\s)^3 - 1.5976*(x-\m)/\s));%
    }
}

%--- Lectures --------------------------------------------------------%
\include{1-Estatistica_Bayesiana}
\include{2-Distribuicoes_Estatisticas}
\include{3-rstanarm_brms}
\include{4-Priors}
\include{5-Priors_Posteriors_Checks}
\include{6-Regressao_Linear}
\include{7-Regressao_Logistica}
\include{7.1-Regressao_Ordinal}
\include{8-Regressao_Poisson}
\include{9-Regressao_Robusta}
\include{10-Modelos_Multiniveis}
\include{11-MCMC}
\include{12-Comparacao_Modelos}

%--- Citations -------------------------------------------------------%

\begingroup
    \AtBeginSection[]{}
    \section{Referências}
    \begin{frame}[allowframebreaks]{Referências}
        \printbibliography
    \end{frame}
\endgroup

%--- Appendix Slides -------------------------------------------------%
\appendix % do not count the following slides for the total number
\section*{Backup Slides}
\begin{frame}[plain, noframenumbering]{Licença}
    \centering
    \vfill
    \Large O texto e as figuras desses slides possuem uma
    \href{https://creativecommons.org/licenses/by-nc-sa/4.0/deed.pt}{Licença
    Creative Commons
    Atribuição-NãoComercial-CompartilhaIgual 4.0 Internacional (CC BY-NC-SA 4.0)}
    \vfill
    \includegraphics[width = 0.2\textwidth]{CC_SA.png}
\end{frame}

\begin{frame}[plain, noframenumbering]{Como citar esse Conteúdo}
  \centering
  \vfill
  \Large \fullcite{storopoli2021estatisticabayesianaR}
  \vfill
\end{frame}

\begin{frame}[plain, noframenumbering, label=appendixmontyhall, fragile]{Simulação Monte Carlo do Problema de Monty Hall em R}
  \begin{lstlisting}[basicstyle=\footnotesize, language=R]
monty <- function(){
  door_car <- sample(1:3L, 1)
  door_pick <- sample(1:3L, 1)
  door_goats <- setdiff(1:3L, door_car)
  if(door_pick == door_car) {
    door_show <- sample(door_goats, 1)
  } else{
    door_show <- setdiff(door_goats, door_pick)
  }
  door_switch <- setdiff(1:3L, c(door_pick, door_show))[1]
  door_pick <- door_switch
}
wins <- 0
ntrials <- 10000
for (i in 1:ntrials){
  wins <- wins + monty()
}
wins / ntrials
    \end{lstlisting}
\end{frame}

\begin{frame}[plain, noframenumbering, fragile]{Simulação Monte Carlo do Problema de Monty Hall em Julia}
  \begin{lstlisting}[basicstyle=\footnotesize, language=Matlab,escapeinside=\{\}]
function monty()
  door_car = rand(1:3)
  door_pick = rand(1:3)
  door_goats = setdiff([1;2;3],door_car)
  door_show = door_pick == door_car ? rand(door_goats) : setdiff(door_goats,door_pick)
  door_switch = setdiff([1;2;3], [door_pick; door_show])[1]
  door_pick = door_switch
  return door_pick == door_car
end
ntrials = 10_000
wins = reduce(+, (monty() for i {$\in$} 1:ntrials))
wins / ntrials
    \end{lstlisting}
\end{frame}

\begin{frame}[plain, noframenumbering, label=appendixnormal]{Como surgiu a distribuição Normal\footnote{Abraham de Moivre em 1738}\footnote{Uma melhor explanação pode ser encontrada \href{http://www.stat.yale.edu/~pollard/Courses/241.fall2014/notes2014/Bin.Normal.pdf}{clidando aqui}}}
  $$
  \begin{aligned}
    \text{Binomial}(n, k) &= \binom{n}{k} p^k (1-p)^{n-k} \\
    n! &\approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n &\text{(Stirling)} \\
    \lim_{n \to \infty} \binom{n}{k} p^k (1-p)^{n-k} &= \frac{1}{\sqrt{2 \pi npq}} e^{-\frac{(k - np)^2}{2npq}}
  \end{aligned}
  $$
  Sabemos que na binomial: $\mathrm{E} = np$ e $\mathrm{Var} = npq$; logo substituindo $\mathrm{E}$ por $\mu$ $\mathrm{Var}$ por $\sigma^2$:
  $$\lim_{n \to \infty} \binom{n}{k} p^k (1-p)^{n-k} = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(k - \mu)^2}{\sigma^2}}$$
\end{frame}

\begin{frame}[plain, noframenumbering, label=appendixscopus]{Buscas Scopus do \texttt{Stan}}
  \begin{vfilleditems}
      \item Uso Geral na Scopus: \texttt{ALL((brms AND burkner) OR (gelman AND hoffman AND stan) OR mc-stan.org OR rstanarm OR pystan OR (rstan AND NOT mit))}
      \item Uso em Comparação com outras Ferramentas Bayesianas: \texttt{REF((gelman AND hoffman AND stan) OR mc-stan.org) AND REF(brms AND burkner) AND REF(pystan) AND REF(rstanarm) AND REF(rstan AND NOT mit)}
      \item Uso das Ferramentas Bayesians: a busca anterior adicionado de \texttt{REF(PyMC3 OR (PyMC* AND fonnesbeck)) AND REF(tensorflow) AND REF(pytorch) AND REF(Keras)}
  \end{vfilleditems}
\end{frame}

\begin{frame}[plain, noframenumbering, label=appendixnqr]{Decomposição QR}
  \footnotesize
  Em Álgebra Linear 101 aprendemos que qualquer matriz
  (até mesmo as retangulares) podem ser decompostas em um produto de duas matrizes:
  \begin{vfilleditems}
    \footnotesize
    \item $\mathbf{Q}$: uma matriz ortogonal (suas colunas são vetores unitários
    ortogonais, \textit{i.e.} $\mathbf{Q}^T = \mathbf{Q}^{-1}$)
    \item $\mathbf{R}$: uma matrix triangular superior
  \end{vfilleditems}
  \footnotesize
  Agora vamos incorporar a decomposição QR no modelo de regressão linear.
  Aqui, usarei o QR "fino" em vez do "gordo", que escala $\mathbf{Q}$ e $\mathbf{R}$
  matrizes por um fator de $\sqrt{n-1}$ onde $n$ é o número de linhas de $\mathbf{X}$.
  Na prática, é melhor implementar a decomposição QR fina, que é preferível à decomposição QR gorda.
  É numericamente mais estável. Matematicamente, a decomposição QR fina é:
  $$
  \begin{aligned}
  \mathbf{X}       &= \mathbf{Q}^* \mathbf{R}^* \\
  \mathbf{Q}^*     &= \mathbf{Q} \cdot \sqrt{n - 1} \\
  \mathbf{R}^*     &= \frac{1}{\sqrt{n - 1}} \cdot \mathbf{R}\\
  \boldsymbol{\mu} &= \alpha + \mathbf{X} \cdot \boldsymbol{\beta} + \sigma \\
                   &= \alpha + \mathbf{Q}^* \cdot \mathbf{R}^* \cdot \boldsymbol{\beta} + \sigma \\
                   &= \alpha + \mathbf{Q}^* \cdot (\mathbf{R}^* \cdot \boldsymbol{\beta}) + \sigma \\
                   &= \alpha + \mathbf{Q}^* \cdot \widetilde{\boldsymbol{\beta}} + \sigma
  \end{aligned}
  $$
\end{frame}

\end{document}