From e9ad646d68be16add4d89d6a6ba0ff40d6dca516 Mon Sep 17 00:00:00 2001 From: Arnaud Legrand <arnaud.legrand@imag.fr> Date: Mon, 27 May 2019 17:13:01 +0200 Subject: [PATCH] Initial import from old repos --- Makefile | 26 + babel_images/coin_new_bayesian.pdf | Bin 0 -> 5279 bytes babel_images/coin_new_frequentist.pdf | Bin 0 -> 5088 bytes babel_images/density_coin_0_0.pdf | Bin 0 -> 5899 bytes babel_images/density_coin_0_1.pdf | Bin 0 -> 9875 bytes babel_images/density_coin_10_13.pdf | Bin 0 -> 12389 bytes babel_images/density_coin_110_140.pdf | Bin 0 -> 38427 bytes babel_images/density_coin_1_1.pdf | Bin 0 -> 9919 bytes babel_images/density_coin_1_2.pdf | Bin 0 -> 10131 bytes babel_images/density_coin_1_3.pdf | Bin 0 -> 10152 bytes babel_images/density_coin_2_3.pdf | Bin 0 -> 10298 bytes babel_images/density_coin_2_4.pdf | Bin 0 -> 10510 bytes babel_images/density_coin_30_35.pdf | Bin 0 -> 17218 bytes babel_images/density_coin_3_4.pdf | Bin 0 -> 10621 bytes babel_images/density_mu_sigma1.pdf | Bin 0 -> 40304 bytes babel_images/density_mu_sigma1_prior.pdf | Bin 0 -> 40195 bytes babel_images/density_mu_sigma1_zoom.pdf | Bin 0 -> 42768 bytes babel_images/density_mu_sigma_2d.pdf | Bin 0 -> 14944 bytes babel_images/density_mu_sigma_2d_1.pdf | Bin 0 -> 6464 bytes babel_images/density_mu_sigma_2d_2.pdf | Bin 0 -> 6775 bytes babel_images/density_mu_sigma_2d_3.pdf | Bin 0 -> 7044 bytes babel_images/density_mu_sigma_2d_4.pdf | Bin 0 -> 7345 bytes babel_images/density_tcoin_0_0.pdf | Bin 0 -> 8428 bytes babel_images/density_tcoin_0_1.pdf | Bin 0 -> 10226 bytes babel_images/density_tcoin_10_13.pdf | Bin 0 -> 12765 bytes babel_images/density_tcoin_110_140.pdf | Bin 0 -> 38852 bytes babel_images/density_tcoin_1_1.pdf | Bin 0 -> 10293 bytes babel_images/density_tcoin_1_2.pdf | Bin 0 -> 10517 bytes babel_images/density_tcoin_1_3.pdf | Bin 0 -> 10518 bytes babel_images/density_tcoin_2_3.pdf | Bin 0 -> 10644 bytes babel_images/density_tcoin_2_4.pdf | Bin 0 -> 10875 bytes babel_images/density_tcoin_30_35.pdf | Bin 0 -> 17607 bytes babel_images/density_tcoin_3_4.pdf | Bin 0 -> 10981 bytes babel_images/direct_gen_density.pdf | Bin 0 -> 6947 bytes babel_images/direct_gen_pdf.pdf | Bin 0 -> 5704 bytes babel_images/direct_gen_qf.pdf | Bin 0 -> 5784 bytes babel_images/direct_gen_rnorm.pdf | Bin 0 -> 5345 bytes babel_images/direct_gen_runif.pdf | Bin 0 -> 5966 bytes babel_images/hist1.pdf | Bin 0 -> 4965 bytes babel_images/hist1_1.pdf | Bin 0 -> 6256 bytes babel_images/hist1_2.pdf | Bin 0 -> 6016 bytes babel_images/hist1_3.pdf | Bin 0 -> 6060 bytes babel_images/hist1_4.pdf | Bin 0 -> 6193 bytes babel_images/stan_convergence.pdf | Bin 0 -> 35902 bytes babel_images/stan_data.pdf | Bin 0 -> 33295 bytes babel_images/stan_hist.pdf | Bin 0 -> 5423 bytes babel_images/stan_samples.pdf | Bin 0 -> 29247 bytes bayesian_statistics_introduction.org | 1337 ++++++++++++++++++++++ img/bayes_gelman.jpg | Bin 0 -> 10487 bytes img/bayes_mcelreath.jpg | Bin 0 -> 9464 bytes img/gelman_264.pdf | Bin 0 -> 19850 bytes img/gelman_277.pdf | Bin 0 -> 2449529 bytes img/stanislaw_ulam.png | Bin 0 -> 118764 bytes 53 files changed, 1363 insertions(+) create mode 100644 Makefile create mode 100644 babel_images/coin_new_bayesian.pdf create mode 100644 babel_images/coin_new_frequentist.pdf create mode 100644 babel_images/density_coin_0_0.pdf create mode 100644 babel_images/density_coin_0_1.pdf create mode 100644 babel_images/density_coin_10_13.pdf create mode 100644 babel_images/density_coin_110_140.pdf create mode 100644 babel_images/density_coin_1_1.pdf create mode 100644 babel_images/density_coin_1_2.pdf create mode 100644 babel_images/density_coin_1_3.pdf create mode 100644 babel_images/density_coin_2_3.pdf create mode 100644 babel_images/density_coin_2_4.pdf create mode 100644 babel_images/density_coin_30_35.pdf create mode 100644 babel_images/density_coin_3_4.pdf create mode 100644 babel_images/density_mu_sigma1.pdf create mode 100644 babel_images/density_mu_sigma1_prior.pdf create mode 100644 babel_images/density_mu_sigma1_zoom.pdf create mode 100644 babel_images/density_mu_sigma_2d.pdf create mode 100644 babel_images/density_mu_sigma_2d_1.pdf create mode 100644 babel_images/density_mu_sigma_2d_2.pdf create mode 100644 babel_images/density_mu_sigma_2d_3.pdf create mode 100644 babel_images/density_mu_sigma_2d_4.pdf create mode 100644 babel_images/density_tcoin_0_0.pdf create mode 100644 babel_images/density_tcoin_0_1.pdf create mode 100644 babel_images/density_tcoin_10_13.pdf create mode 100644 babel_images/density_tcoin_110_140.pdf create mode 100644 babel_images/density_tcoin_1_1.pdf create mode 100644 babel_images/density_tcoin_1_2.pdf create mode 100644 babel_images/density_tcoin_1_3.pdf create mode 100644 babel_images/density_tcoin_2_3.pdf create mode 100644 babel_images/density_tcoin_2_4.pdf create mode 100644 babel_images/density_tcoin_30_35.pdf create mode 100644 babel_images/density_tcoin_3_4.pdf create mode 100644 babel_images/direct_gen_density.pdf create mode 100644 babel_images/direct_gen_pdf.pdf create mode 100644 babel_images/direct_gen_qf.pdf create mode 100644 babel_images/direct_gen_rnorm.pdf create mode 100644 babel_images/direct_gen_runif.pdf create mode 100644 babel_images/hist1.pdf create mode 100644 babel_images/hist1_1.pdf create mode 100644 babel_images/hist1_2.pdf create mode 100644 babel_images/hist1_3.pdf create mode 100644 babel_images/hist1_4.pdf create mode 100644 babel_images/stan_convergence.pdf create mode 100644 babel_images/stan_data.pdf create mode 100644 babel_images/stan_hist.pdf create mode 100644 babel_images/stan_samples.pdf create mode 100644 bayesian_statistics_introduction.org create mode 100644 img/bayes_gelman.jpg create mode 100644 img/bayes_mcelreath.jpg create mode 100644 img/gelman_264.pdf create mode 100644 img/gelman_277.pdf create mode 100644 img/stanislaw_ulam.png diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..cbb85a9 --- /dev/null +++ b/Makefile @@ -0,0 +1,26 @@ +include LaTeX.mk + +LU_FLAVORS=LUALATEX +LUALATEX_OPTIONS=-shell-escape + +all: + export PYTHONIOENCODING=utf-8 + make bayesian_statistics_introduction.tex + make bayesian_statistics_introduction.pdf +# make fix_figures +# make bayesian_statistics_introduction.pdf + +%.tex: %.org + emacs -batch --eval "(setq enable-local-eval t)" --eval "(setq enable-local-variables t)" $< --funcall org-beamer-export-to-latex + mv $@ $@.bak + echo '\\def\\raggedright{}' > $@ + cat $@.bak >> $@ + rm $@.bak + sed -i -e 's/usepackage{graphicx}/usepackage{figlatex}/' -e 's/\\usepackage{grffile}//' $@ + + +include LaTeX.mk + +clean:: + $(RM) $(BIB_SLAVES) + diff --git a/babel_images/coin_new_bayesian.pdf b/babel_images/coin_new_bayesian.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2920c843da82fe2fb54f4d04ecaeca774c483d74 GIT binary patch literal 5279 zcmY!laB<T$)HC5yZG6=Dxbb}B<(MKaLj?nc{G=={8yhZt=c3falFa-(m&B4(1q~M~ 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0000000..dd09539 --- /dev/null +++ b/bayesian_statistics_introduction.org @@ -0,0 +1,1337 @@ +# -*- coding: utf-8 -*- +# -*- mode: org -*- +#+Title: Introduction to Bayesian Statistics +#+Author: Arnaud Legrand\newline Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP +#+DATE: POLARIS days, May 2019 +#+LANGUAGE: en +#+STARTUP: beamer indent inlineimages logdrawer +#+TAGS: noexport(n) + +#+PROPERTY: header-args :session :eval never-export :exports both +#+DRAWERS: latex_headers + +:latex_headers: +#+LaTeX_CLASS: beamer +#+LATEX_CLASS_OPTIONS: [10pt,presentation,xcolor={usenames,dvipsnames,svgnames,table}] +#+OPTIONS: H:2 num:t toc:nil \n:nil @:t ::t |:t ^:nil -:t f:t *:t <:t +#+LATEX_COMPILER: lualatex +#+LATEX_HEADER: \usedescriptionitemofwidthas{bl} +#+LATEX_HEADER: \usepackage[T1]{fontenc} +#+LATEX_HEADER: \usepackage[utf8]{inputenc} +#+LATEX_HEADER: \usepackage{figlatex} +#+LATEX_HEADER: \usepackage[french]{babel} +#+LATEX_HEADER: %\usepackage{DejaVuSansMono} +#+LATEX_HEADER: \usepackage{ifthen,amsmath,amstext,gensymb,amssymb} +#+LATEX_HEADER: \usepackage{boxedminipage,xspace,multicol} +#+LATEX_HEADER: %%%%%%%%% Begin of Beamer Layout %%%%%%%%%%%%% +#+LATEX_HEADER: \ProcessOptionsBeamer +#+LATEX_HEADER: \usetheme[numbering=fraction,titleformat=smallcaps,progressbar=frametitle]{metropolis} +#+LATEX_HEADER: \usepackage{fontawesome} +#+LATEX_HEADER: \usecolortheme[named=BrickRed]{structure} +#+LATEX_HEADER: %%%%%%%%% End of Beamer Layout %%%%%%%%%%%%% +#+LATEX_HEADER: \usepackage{verbments} +#+LATEX_HEADER: \usepackage{xcolor} +#+LATEX_HEADER: \usepackage{color} +#+LATEX_HEADER: \usepackage{url} \urlstyle{sf} +#+LATEX_HEADER: \let\alert=\structure % to make sure the org * * works of tools +#+LATEX_HEADER: %\let\tmptableofcontents=\tableofcontents +#+LATEX_HEADER: %\def\tableofcontents{} +#+LATEX_HEADER: \usepackage[normalem]{ulem} +#+LATEX_HEADER: \usepackage{color,soul} +#+LATEX_HEADER: \definecolor{lightorange}{rgb}{1,.9,.7} +#+LATEX_HEADER: \sethlcolor{lightorange} +#+LATEX_HEADER: \definecolor{lightgreen}{rgb}{.7,.9,.7} +#+LATEX_HEADER: \let\hrefold=\href +#+LATEX_HEADER: \renewcommand{\href}[2]{\hrefold{#1}{\SoulColor{lightorange}\hl{#2}}} +#+LATEX_HEADER: % \renewcommand{\uline}[1]{\SoulColor{lightorange}\hl{#1}} +#+LATEX_HEADER: \renewcommand{\emph}[1]{\SoulColor{lightorange}\hl{#1}} +#+LATEX_HEADER: \makeatletter +#+LATEX_HEADER: \newcommand\SoulColor[1]{% +#+LATEX_HEADER: \sethlcolor{#1}% +#+LATEX_HEADER: \let\set@color\beamerorig@set@color% +#+LATEX_HEADER: \let\reset@color\beamerorig@reset@color} +#+LATEX_HEADER: \makeatother +#+LATEX_HEADER: \let\oldtexttt=\texttt +#+LATEX_HEADER: % \renewcommand\texttt[1]{\SoulColor{lightgreen}\hl{\tt#1}} +#+LATEX_HEADER: % \renewcommand\alert[1]{\SoulColor{lightgreen}\hl{#1}} +#+LATEX_HEADER: % \AtBeginSection{\begin{frame}{Outline}\tableofcontents\end{frame}} +#+LATEX_HEADER: \graphicspath{{fig/}} +#+LATEX_HEADER: \usepackage{tikzsymbols} +#+LATEX_HEADER: \def\smiley{\Smiley[1][green!80!white]} +#+LATEX_HEADER: \def\frowny{\Sadey[1][red!80!white]} +#+LATEX_HEADER: \def\winkey{\Winkey[1][yellow]} + +#+BEGIN_EXPORT latex + \newcommand{\myfbox}[2][gray!20]{\bgroup\scalebox{.7}{\colorbox{#1}{{\vphantom{pS}#2}}}\egroup} % \fbox + %\def\myfbox#1{#1} % \fbox + \def\HPC{\myfbox[gray!40]{HPC}} + \def\NET{\myfbox[gray!40]{Network}} + \def\SG{\myfbox[gray!40]{Smart Grids}} + \def\ECO{\myfbox[gray!40]{Economics}} + \def\PRIV{\myfbox[gray!40]{Privacy}} + \def\TRACING{\myfbox[red!20]{Tracing}} + \def\SIM{\myfbox[green!20]{Simulation}} + \def\VIZ{\myfbox[red!40]{Visualization}} + \def\MODELING{\myfbox[green!40]{Stochastic Models}} + \def\OPT{\myfbox[blue!20]{Optimization}} + \def\GT{\myfbox[blue!40]{Game Theory}} +#+END_EXPORT + + +#+BEGIN_EXPORT latex +\def\changefont#1{% + \setbeamertemplate{itemize/enumerate body begin}{#1} + \setbeamertemplate{itemize/enumerate subbody begin}{#1} + #1} +\makeatletter +\newcommand{\verbatimfont}[1]{\renewcommand{\verbatim@font}{\ttfamily#1}} +\makeatother +\verbatimfont{\scriptsize}%small +\let\endmintedbak=\endminted +\def\endminted{\endmintedbak\vspace{-1cm}} +#+END_EXPORT + +#+BEGIN_EXPORT latex +\newcommand{\Norm}{\ensuremath{\mathcal{N}}\xspace} +\newcommand{\Unif}{\ensuremath{\mathcal{U}}\xspace} +\newcommand{\Triang}{\ensuremath{\mathcal{T}}\xspace} +\newcommand{\Exp}{\ensuremath{\mathcal{E}}\xspace} +\newcommand{\Bernouilli}{\ensuremath{\mathcal{B}}\xspace} +\newcommand{\Like}{\ensuremath{\mathcal{L}}\xspace} +\newcommand{\Model}{\ensuremath{\mathcal{M}}\xspace} +\newcommand{\E}{\ensuremath{\mathbb{E}}\xspace} +\def\T{\ensuremath{\theta}\xspace} +\def\Th{\ensuremath{\hat{\theta}}\xspace} +\def\Tt{\ensuremath{\tilde{\theta}}\xspace} +\def\Y{\ensuremath{y}\xspace} +\def\Yh{\ensuremath{\hat{y}}\xspace} +\def\Yt{\ensuremath{\tilde{y}}\xspace} +\let\epsilon=\varepsilon +\let\leq=\leqslant +\let\geq=\geqslant +#+END_EXPORT +:end: + +# https://cran.r-project.org/web/packages/plot3D/vignettes/plot3D.pdf +# http://htmlpreview.github.io/?https://github.com/AckerDWM/gg3D/blob/master/gg3D-vignette.html + +# http://bechtel.colorado.edu/~bracken/tutorials/stan/stan-tutorial.pdf +# http://jakewestfall.org/misc/SorensenEtAl.pdf +# https://github.com/AllenDowney/BayesMadeSimple + +# https://github.com/bob-carpenter/prob-stats + +#+BEGIN_EXPORT latex +#+END_EXPORT + +** Goal +Given a some observations, how to generate "similar" values? +- +GANs ? No, not today+ +- How to leverage your knowledge about the system? +- How to check whether it is reasonable or not? + +*** This talk: +1. Brief introduction to Bayesian statistics. + # Bien comprendre le statut des variables que l'on manipule + # (paramètres, observations, prédicteurs) +2. Brief introduction to Bayesian sampling +3. Brief presentation of STAN + +* Random Thoughts :noexport: +** Progression +- Bayes Theorem provides a way to get from P(A|B) to P(B|A) +- Illustration Cookies (fully discrete) +- Bayesian coin (discrete / continuous), + - allows to easily show how unimportant the prior is. +- Linear regression (continuous / continuous) mais avec dependance + entre les estimateurs + - allows to easily inject constraints on estimates through the prior + (e.g., positive coefficient) + - max sans grande importance, distribution bien plus utile +** Sampling +- Prior utile +- Complex models, hierarchical +- How to sample + +* Bayesian Statistics +** Useful R plotting functions :noexport: + +#+begin_src R :results output :session *R* :exports none +plot_histogram <- function (data, xmin=NA, xmax=NA, ymax=NA, ypos=.1, binwidth=1) { + require(ggplot2) + set.seed(42); + dataY=data.frame(Y=data) + p = ggplot(dataY, aes(x=Y)) + + geom_histogram(aes(y=..density..), binwidth=binwidth, boundary=0, + color="black", fill="gray") + + geom_jitter(aes(y=ypos),height=ypos/2) + + ylim(0,ymax) + + theme_bw(); # bins=10 + if(!is.na(xmin) | !is.na(xmax)) { + p = p + coord_cartesian(xlim=c(xmin,xmax),ylim=c(0,ymax)) + } else { p = p + coord_cartesian(ylim=c(0,ymax)) } + return(p); +} + +likelihood_norm <- function (mu, sigma, X=data) { + return(prod(dnorm(X,mean=mu,sd=sigma)*(1:length(X)))) +} +## likelihood_norm <- function (mu, sigma, X=data) { +## d = 1; +## for(x in X) { +## d = d*1/(sigma*sqrt(2*pi))*exp(-1/2*((x-mu)/sigma)**2); +## } +## return(d); +## } + +likelihood_3d <- function(data, xmin=0, xmax=20, ymin=0, ymax=5, length=50, + likelihood_function = likelihood_norm ) { + x <- seq(xmin, xmax, length=length) + y <- seq(ymin, ymax, length=length) + xy <- expand.grid(x,y) + names(xy)=c("x","y"); + xy$z=0; + for(i in 1:nrow(xy)) { + xy[i,]$z=likelihood_norm(xy[i,]$x,xy[i,]$y,data) + } + if(prod(is.na(xy$z))) { xy[is.na(xy$z),]$z <- 0 } + return(xy); +} + +plot_likelihood_3d <- function(l3d,xlab="$\\mu$",ylab="$\\sigma$") { + require(lattice) + require(latex2exp) + ## l3d = likelihood_3d(data, xmin=xmin, xmax=xmax, ymin=ymin, ymax=ymax, length=length, + ## likelihood_function=likelihood_function); + ## print(l3d); + theseCol=colorRampPalette(c("yellow", "red"))(150) + wireframe(z~x*y, data = l3d, + xlab=TeX(xlab), ylab=TeX(ylab), zlab = "", + colorkey=F,col.regions=theseCol,drape=T, + scales = list( arrows=FALSE, z = list(tick.number=0)), + screen = list(z=40,x=-50,y=0), + par.settings = list(axis.line = list(col = "transparent")) + ) + # xlab=expression(mu), ylab=expression(sigma) +} + +plot_likelihood_2d <- function(l3d,labels=data.frame(), + background=T,xlab="$\\mu$",ylab="$\\sigma$") { + require(ggplot2); + require(latex2exp) + require(ggrepel); + set.seed(42); + + dx = unique(df$x); + dx = mean(dx[-1]-dx[-length(dx)]) + dy = unique(df$y); + dy = mean(dy[-1]-dy[-length(dy)]) + + p = ggplot(l3d,aes(x=x,y=y,fill=z)) + + theme_bw() + xlab(TeX(xlab)) + ylab(TeX(ylab)) + + theme(legend.position="none") + + xlim(min(l3d$x),max(l3d$x)) + + ylim(min(l3d$y),max(l3d$y)) + + scale_fill_gradient(low="yellow", high="red"); + + if(background) { + p = p + geom_tile(width=dx,height=dy) ; + } + if(nrow(labels)>0) { + p = p + geom_point(data=labels) + + geom_label_repel(data=labels,aes(label=round(z,2)),fill="white", + box.padding=T,size=4, + force = 10); + } + return(p); +} +#+end_src + +#+RESULTS: + +** Background on Bayesian Statistics +- Model :: we assume $y \sim \Model(\theta,x)$ + - *$\theta$*: Model *parameters* + - *$y$*: Dependent *data* (response) + - *$x$*: Independent data (covariates/\alert{predictors}/constants) + +- Examples :: + - $y \sim \Norm(\mu,\sigma)$ + - $y \sim x^2 + \Unif(\alpha,\beta)$ + - $y \sim \Norm(\alpha x + \beta, \sigma)$ + - $y \sim \Norm(\alpha \log(x) + \beta, \alpha' x + \beta')$ + +_Everyone_: Model data as random +** Bayes Rule +- Notation :: + - $p(A)$ = probability that $A$ occurs + - $p(A,B)$ = probability that $A$ and $B$ occurs + - $p(A|B)$ = probability that $A$ occurs, given that $B$ occurs \pause +- Conjunction rule :: + - $p(A,B) = p(A|B)p(B)$ + - $p(B,A) = p(B|A)p(A)$ \pause +- Bayes rule :: Equate and divide by $p(B)$ + $$\boxed{p(A|B) = \frac{p(B|A)p(A)}{p(B)}}$$ +** Background on Bayesian Statistics + +_Bayesians_: Data is fixed (observed), model parameters as random\vspace{-2em} + +#+BEGIN_EXPORT latex +\begin{align*} + p(\theta,y,x) & = p(y,\theta,x)\\ + p(\theta|y,x)p(y,x) & = p(y|\theta,x)p(\theta,x) +\end{align*} +\begin{align*} + \text{Hence } \alert{p(\theta|y,x)} & = \frac{p(y|\theta,x)p(\theta,x)}{p(y,x)} = \frac{p(y|\theta,x)p(\theta)p(x)}{p(y,x)}\\ + & \alert{\propto p(y|\theta,x)p(\theta)} \text{~~ ($y$, and $x$ are fixed for a given data set)} +\end{align*} +#+END_EXPORT +\pause\vspace{-1em} + +- Bayes rule :: + $\boxed{\underbrace{p(\theta|y,x)}_{\text{\alert{Posterior}}} \propto \underbrace{p(y|\theta,x)}_{\text{\alert{Likelihood}}}\underbrace{p(\theta,x)}_{\text{\alert{Prior}}}}$ + assuming $y \sim \Model(\theta,x)$ + + - *Posterior*: The answer, probability distributions of parameters + - *Likelihood*: + #+LaTeX: \hbox{A (model specific) computable function of the parameters\hspace{-1cm}} + - *Prior*: "Initial guess", existing knowledge of the system + +The key to building Bayesian models is specifying the likelihood +function, same as frequentist. +* A € problem +** R code :noexport: + #+begin_src R :results output :session *R* :exports none + set.seed(42); + gen_binom_fixed = function(n1,n2) { + set.seed(42); + return(sample(c(rep(0,n1),rep(1,n2)),n1+n2)) + } + Comb = function(n,k) { + if(n==0 || k==0) { return(1);} + return(prod(((n-k+1):n)/(1:k))) + } + bernouilli_likelihood = function(pi,n1,n2,triangular_prior=F) { + prior = 1 + if(triangular_prior) { + prior = (2-4*abs(pi-1/2)); + prior = prior*((1+n2)/(n1+n2+1)); #Hack the normalisation for triangular prior + } + return(prior*pi^n2*(1-pi)^n1*Comb(n1+n2,n1)*(n1+n2+1)) + #*factorial(n1+n2)/factorial(n1)/factorial(n2) + } + ## bernouilli_likelihood(.4,140,110) + ## bernouilli_likelihood(.44,140,110) + ## bernouilli_likelihood(.5,140,110) + ## bernouilli_likelihood(.56,140,110) + ## bernouilli_likelihood(.44,140,110)/ + ## bernouilli_likelihood(.44,140,110,triangular_prior=T) + ## print("") + ## bernouilli_likelihood(.4,0,0) + #+end_src + + #+RESULTS: + + #+begin_src R :results output graphics :file babel_images/density_coin_0_0.pdf :exports results :width 6 :height 2.5 :session *R* + library(ggplot2) + require(gridExtra) + + plot_coin = function(n1,n2, prior=T, likelihood = T, triangular_prior=F) { +# if(n1==0 && n2==0) {n1=1; n2=1;} + dataY=data.frame(Y=gen_binom_fixed(n1,n2)); + dataY$ypos=(.25+.5*runif(nrow(dataY)))*nrow(dataY)/2; + dataY$xpos=(.9*dataY$Y+.1*runif(nrow(dataY))); + p1 = ggplot(dataY, aes(x=Y)) + + geom_histogram(binwidth=.1, boundary=0, color="black", fill="gray") + + geom_point(aes(y=ypos,x=xpos),alpha=.4) + + theme_bw() + xlim(0,1)# + ylim(0,5) +# plot_histogram(gen_binom_fixed(n1,n2),xmin=0,xmax=1,ymax=5, ypos=.3, binwidth=.1) + p2 = ggplot(data=data.frame()) + ylim(0,15) + theme_bw() + xlim(0,1) + xlab(expression(pi)) + ylab("density"); + if(prior) { + p2 = p2 + geom_area(data=data.frame(Y=c(0,1)), stat = "function", + fun = bernouilli_likelihood, fill="blue", alpha=.2, + args=list(n1=0,n2=0,triangular_prior=triangular_prior)); + } + if(likelihood) { + p2 = p2 + geom_area(data=data.frame(Y=c(0,1)), stat = "function", + fun = bernouilli_likelihood, fill="red", alpha=.4, + args=list(n1=n1,n2=n2,triangular_prior=triangular_prior)); + } + return(grid.arrange(p1, p2, ncol=2)); + } +# plot_coin(0,0, prior=T, likelihood=F) + plot_coin(0,0,likelihood=F) + #+end_src + + #+RESULTS: + [[file:babel_images/density_coin_0_0.pdf]] + + #+name: coin_values + | n1 | n2 | + |-----+-----| + | 0 | 1 | + | 1 | 1 | + | 1 | 2 | + | 1 | 3 | + | 2 | 3 | + | 2 | 4 | + | 3 | 4 | + | 10 | 13 | + | 30 | 35 | + | 110 | 140 | + + #+begin_src R :results output :session *R* :exports both :var coin_values=coin_values + for(i in 1:nrow(coin_values)) { + n1 = coin_values[i,]$n1; + n2 = coin_values[i,]$n2; + pdf(file=paste0("babel_images/density_coin_",n1,"_",n2,".pdf"),width=6,height=2.5); + plot(plot_coin(n1,n2)); + dev.off(); + } + pdf(file=paste0("babel_images/density_tcoin_0_0.pdf"),width=6,height=2.5); + plot(plot_coin(0,0,triangular_prior=T,likelihood=F)); + dev.off(); + for(i in 1:nrow(coin_values)) { + n1 = coin_values[i,]$n1; + n2 = coin_values[i,]$n2; + pdf(file=paste0("babel_images/density_tcoin_",n1,"_",n2,".pdf"),width=6,height=2.5); + plot(plot_coin(n1,n2,triangular_prior=T)); + dev.off(); + } + #+end_src + + #+RESULTS: + : png + : 2 + +#+begin_src R :results output :session *R* :exports both +Comb(250,110)/2**250 +#+end_src + +#+RESULTS: +: [1] 0.008357182 + +#+begin_src R :results output :session *R* :exports none +s=0 +for(k in 1:110) { + s = s+ Comb(250,k)/2**250 +} +s +#+end_src + +#+RESULTS: +: [1] 0.03321058 + +** Head and Tail +#+BEGIN_QUOTE +When spun on edge 250 times, a Belgian 1€ coin came up heads 140 +times and tails 110. It looks very suspicious to me. If the coin were +unbiased, the chance of getting a result as extreme as that would be +less than 7%. + +\flushright\scriptsize -- From "The Gardian" quoted by MacKay @@beamer:\\@@ +in *Information Theory, Inference and Learning Algorithms* +#+END_QUOTE + +#+LaTeX: \begin{columns}\begin{column}{.35\linewidth} + # #+ATTR_BEAMER: :overlay <+-> + - Model: $Y \sim \Bernouilli(\pi)$ + - + #+LaTeX: Data:~\rlap{\hbox{\scalebox{.75}{$y=1,0,1,1,0,0,1,1,1,\dots$}}} + - + #+LaTeX: \only<1>{$p(y|\pi=1/2)$\\ \hbox{~~$=\frac{(140+110)!}{110!140!}.(\frac{1}{2})^{110}.(\frac{1}{2})^{140}$} \hbox{$~~\approx 0.00835$}}% + #+LaTeX: \only<2>{$p(y|\pi\le1/2)$\\ \hbox{~~$=\sum_{k\le110}\frac{250!}{k!(250-k)!}.\frac{1}{2^{250}}$} \hbox{$~~\approx 0.033$}}% + #+LaTeX: \only<3->{Prior: $\pi \sim \only<3-14>{\Unif(0,1)}\only<15->{\Triang(0,1)}$\\~\\} +#+LaTeX: \end{column}\begin{column}{.65\linewidth} + #+BEGIN_EXPORT latex + \phantom{\includegraphics<+>[width=\linewidth]{babel_images/density_coin_0_0.pdf}}% + \phantom{\includegraphics<+>[width=\linewidth]{babel_images/density_coin_0_0.pdf}}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_0_0.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_0_1.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_1_1.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_1_2.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_1_3.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_2_3.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_2_4.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_3_4.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_10_13.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_30_35.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_110_140.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_0_0.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_0_0.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_0_1.pdf}% +% \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_1_1.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_1_2.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_1_3.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_2_3.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_2_4.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_3_4.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_10_13.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_30_35.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_tcoin_110_140.pdf}% + \includegraphics<+>[width=\linewidth]{babel_images/density_coin_110_140.pdf}% + #+END_EXPORT +#+LaTeX: \end{column}\end{columns} + #+BEGIN_EXPORT latex + \begin{equation*} + \uncover<3->{p(\pi|y) = \frac{p(y|\pi)\cdot p(\pi)}{p(y)}= \frac{(1-\pi)^{n_0}\pi^{n_1} \cdot \only<3-14>{1}\only<15->{(2-4|\pi-0.5|)}}{\only<3-14>{n_0!n_1!/(n_0+n_1+1)!}\only<15->{\text{some normalization}}}} + \end{equation*} + #+END_EXPORT + +* A Simple Gaussian Model +** Initial Belief and First Observations +#+LaTeX: \begin{columns}\begin{column}{.6\linewidth} + # #+ATTR_BEAMER: :overlay <+-> + - Model: $Y \sim \Norm(\mu,\sigma)$ + - Prior: $\mu \sim \Unif(0,20)$ and $\sigma \sim \Unif(0,5)$ +#+LaTeX: \end{column}\begin{column}{.4\linewidth} + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1_prior.pdf]] +#+LaTeX: \end{column}\end{columns}\pause + +#+begin_src R :results output :session *R* :exports both +set.seed(162); +n=20; mu=12.5; sigma=1.6; +Y=rnorm(n, mean=mu, sd=sigma); +Y +#+end_src + +#+RESULTS: +: [1] 13.899247 12.951346 12.164091 10.869858 13.075777 12.552552 15.446823 +: [8] 11.920264 12.849875 9.367122 12.083848 13.852930 12.740590 9.674321 +: [15] 11.489182 12.195024 13.946985 9.220992 11.821921 9.347013 +*** Graphs :noexport: +#+begin_src R :results output graphics :file babel_images/density_mu_sigma1_prior.pdf :exports both :width 7 :height 7 :session *R* +l3d = likelihood_3d(c(-120,-120)); +xrange = max(l3d$x) - min(l3d$x); +yrange = max(l3d$y) - min(l3d$y); +## l3d$z = l3d$z + 1/(xrange*yrange) +plot_likelihood_3d(l3d) +#+end_src + +#+RESULTS: +[[file:babel_images/density_mu_sigma1_prior.pdf]] + +** Likelihood for This Model +Model: $Y \sim \Norm(\mu,\sigma)$, hence $p(y|\mu,\sigma) = + \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{y-\mu}{2}\right)^2\right)$ + +Therefore $\displaystyle p(\mu,\sigma|y) \propto + \prod_{i=1}^n + \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{y_i-\mu}{2}\right)^2\right)\textcolor{gray}{.\frac{1}{100}}$ +** Exploiting information (Normal model) +#+begin_src R :results output :session *R* :exports results +print(paste("Mean:", mean(Y))) +print(paste("Standard Deviation:", sd(Y))) +#+end_src + +#+RESULTS: +: [1] "Mean: 12.07348806679" +: [1] "Standard Deviation: 1.70127707382769" + +\medskip + +#+LaTeX: \begin{columns}\begin{column}[t]{.5\linewidth}\centering + Distribution of observations $Y$ +#+Beamer: \only<1>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/hist1.pdf]] +#+Beamer: }\only<2>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/hist1_1.pdf]] +#+Beamer: }\only<3>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/hist1_2.pdf]] +#+Beamer: }\only<4>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/hist1_3.pdf]] +#+Beamer: }\only<5>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/hist1_4.pdf]] +#+Beamer: }\only<6->{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/hist1.pdf]] +#+Beamer: } + +#+LaTeX: \end{column}\begin{column}[t]{.5\linewidth}\centering + Posterior distribution + #+LaTeX: \uncover<8>{(Zoom)} + #+Beamer: \only<2>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma_2d_1.pdf]] + #+Beamer: }\only<3>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma_2d_2.pdf]] + #+Beamer: }\only<4>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma_2d_3.pdf]] + #+Beamer: }\only<5>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma_2d_4.pdf]] + #+Beamer: }\only<6>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma_2d.pdf]] + #+Beamer: }\only<7>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1.pdf]] + #+Beamer: }\only<8>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1_zoom.pdf]] + #+Beamer: } +#+LaTeX: \end{column}\end{columns}\pause + +*** Graphs :noexport: +#+begin_src R :results output graphics :file babel_images/hist1.pdf :exports results :width 4 :height 4 :session *R* +p = plot_histogram(Y,xmin=0,xmax=20,ymax=.5); +p +#+end_src + +#+RESULTS: +[[file:babel_images/hist1.pdf]] + +#+begin_src R :results output :session *R* :exports both +l3d = likelihood_3d(data, xmin=0, xmax=20, ymin=0, ymax=5, length=51); +l3d_examples = with(l3d, l3d[(x==5.2 & y==4),]); +l3d_examples = rbind(l3d_examples, with(l3d, l3d[(x==16 & y==.5),])); +l3d_examples = rbind(l3d_examples, with(l3d, l3d[(x==12 & y==.8),])); +l3d_examples = rbind(l3d_examples, with(l3d, l3d[(x==12 & y==2),])); +for(i in 1:nrow(l3d_examples)) { + pdf(file=paste0("babel_images/density_mu_sigma_2d_",i,".pdf"),width=4,height=4); + plot(plot_likelihood_2d(l3d,l3d_examples[1:i,],background=F)); + dev.off(); + pdf(file=paste0("babel_images/hist1_",i,".pdf"),width=4,height=4); + p = plot_histogram(Y,xmin=0,xmax=20,ymax=.8); + p = p + geom_area(data=data.frame(Y=c(0,20)), stat = "function", + fun = dnorm, fill="red", alpha=.4, + args=list(mean=l3d_examples[i,]$x, sd=l3d_examples[i,]$y)); + plot(p); + dev.off(); +} +#+end_src + +#+RESULTS: + +#+begin_src R :results output graphics :file babel_images/density_mu_sigma_2d.pdf :exports both :width 4 :height 4 :session *R* +plot_likelihood_2d(l3d,l3d_examples); +#+end_src + +#+RESULTS: +[[file:babel_images/density_mu_sigma_2d.pdf]] + +#+begin_src R :results output graphics :file babel_images/density_mu_sigma1.pdf :exports both :width 7 :height 7 :session *R* +plot_likelihood_3d(likelihood_3d(Y)) +#+end_src + +#+RESULTS: +[[file:babel_images/density_mu_sigma1.pdf]] + +#+begin_src R :results output graphics :file babel_images/density_mu_sigma1_zoom.pdf :exports both :width 7 :height 7 :session *R* +plot_likelihood_3d(likelihood_3d(data, xmin=10, xmax=14, ymin=0, ymax=4, length=50)) +#+end_src + +#+RESULTS: +[[file:babel_images/density_mu_sigma1_zoom.pdf]] + +#+begin_src shell :results output :exports both +for i in babel_images/density_mu_sigma*.pdf ; do + pdfcrop $i $i ; +done +#+end_src + +#+RESULTS: +#+begin_example +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma1.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma1_prior.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma1_zoom.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma_2d.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma_2d_1.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma_2d_2.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma_2d_3.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/density_mu_sigma_2d_4.pdf'. +#+end_example +** Single point estimate (Normal model) +#+begin_src R :results output :session *R* :exports results +print(paste("Mean:", mean(Y))) +print(paste("Standard Deviation:", sd(Y))) +#+end_src + +#+RESULTS: +: [1] "Mean: 12.07348806679" +: [1] "Standard Deviation: 1.70127707382769" + + $$p(\mu,\sigma|y) \propto \prod_{i=1}^n + \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{y_i-\mu}{2}\right)^2\right)\textcolor{gray}{.\frac{1}{100}}$$ + +- /Machine Learning/: Maximum Likelihood $|y$ + - $\mu_{MLE} = \frac{1}{n} \sum_{i=1}^n y_i$ + - $\sigma_{MLE} = \sqrt{\frac{1}{n} \sum_{i=1}^n (y_i-\mu_{MLE})^2}$ +- /Frequentist/: ensure $\E[\mu_{F}] = \mu$ and $\E[\sigma_{F}^2] = \sigma^2$ + - $\mu_{F} = \frac{1}{n} \sum_{i=1}^n y_i$ + - $\sigma_{F} = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (y_i-\mu_{F})^2}$ +- /Bayesian/: sample the posterior +\vspace{4cm} +** TODO Generating new data +#+begin_src R :results output :session *R* :exports none +set.seed(57) +n = length(Y) +mu_f = mean(Y) +sigma_f = sd(Y) +Y_new=rnorm(10000, mean=mu, sd=sigma); # True distribution +Y_new_f=rnorm(10000, mean=mu_f, sd=sigma_f); # Distribution using a fixed estimation +## Inspire from chap. 14, p.357 of BDA3 by Gelman et al. for this +tdf = n-1 +Y_new_b=mu_f + rt(10000, df = tdf)*(tdf-2)/tdf*sqrt(sigma_f^2*(1+1)); # Bayesian posterior +## Y_new_b=rnorm(10000, mean=Y_new_b, sd=sqrt(31/20*sigma_f)); +Y_df = data.frame(new=Y_new, new_f=Y_new_f, new_b=Y_new_b); +summary(Y_df) +sd(Y_df$new) +sd(Y_df$new_f) +sd(Y_df$new_b) +#+end_src + +#+RESULTS: +#+begin_example + new new_f new_b + Min. : 6.65 Min. : 6.119 Min. : 2.348 + 1st Qu.:11.40 1st Qu.:10.930 1st Qu.:10.600 + Median :12.49 Median :12.058 Median :12.071 + Mean :12.50 Mean :12.071 Mean :12.069 + 3rd Qu.:13.63 3rd Qu.:13.202 3rd Qu.:13.513 + Max. :18.17 Max. :17.950 Max. :21.581 +[1] 1.609831 +[1] 1.698424 +[1] 2.263807 +#+end_example + +- \T: unknown parameter ($\mu=$ src_R[:exports results]{mu} + {{{results(=12.5=)}}}, $\sigma=$ src_R[:exports results]{sigma} {{{results(=1.6=)}}}) +- \Y: observation +- \Th: single point estimate of \T ($\mu\approx$ src_R[:exports results]{round(mu_f,2)} + {{{results(=12.07=)}}}, $\sigma\approx$ src_R[:exports results]{round(sigma_f,2)} {{{results(=1.7=)}}}) +- \Yt: future observations\bigskip + +#+RESULTS: + +#+LaTeX: \begin{columns}\begin{column}[t]{.5\linewidth}\centering +Generating \Yt from \Th +#+begin_src R :results output graphics :file babel_images/coin_new_frequentist.pdf :exports results :width 6 :height 4 :session *R* +library(ggplot2) +ggplot(data=Y_df) + theme_bw() + + geom_histogram(aes(x=new), binwidth=.5, boundary=0, color="black", fill="gray") + + geom_histogram(aes(x=new_f), binwidth=.5, boundary=0, color="black", fill="red", alpha=.4) +#+end_src + +#+RESULTS: +[[file:babel_images/coin_new_frequentist.pdf]] + +(does not account for the uncertainty on \Th) +#+LaTeX: \end{column}\begin{column}[t]{.5\linewidth}\centering\pause +Generating \Yt from many $\Tt|y$ +#+begin_src R :results output graphics :file babel_images/coin_new_bayesian.pdf :exports results :width 6 :height 4 :session *R* +ggplot(data=Y_df) + theme_bw() + + geom_histogram(aes(x=new), binwidth=.5, boundary=0, color="black", fill="gray") + + geom_histogram(aes(x=new_b), binwidth=.5, boundary=0, color="black", fill="blue", alpha=.4) +#+end_src + +#+RESULTS: +[[file:babel_images/coin_new_bayesian.pdf]] + +Noise on $y$ + uncertainty on $\theta$ +#+LaTeX: \end{column}\end{columns} +** TODO What about a different Prior? :noexport: +#+LaTeX: \begin{columns}\begin{column}[t]{.5\linewidth}\centering + \only<1>{Uniform Prior}\only<2>{Non-niform Prior} +#+Beamer: \only<1>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1_prior.pdf]] +#+Beamer: }\only<2>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1_prior.pdf]] +#+Beamer: } + +#+LaTeX: \end{column}\begin{column}[t]{.5\linewidth}\centering + Posterior + #+Beamer: \only<1>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1.pdf]] + #+Beamer: }\only<2>{% + #+ATTR_LATEX: :width \linewidth :center nil + [[file:babel_images/density_mu_sigma1_zoom.pdf]] + #+Beamer: } +#+LaTeX: \end{column}\end{columns}\pause + +*** Graphs :noexport: +#+begin_src R :results output graphics :file babel_images/density_mu_sigma1.pdf :exports both :width 7 :height 7 :session *R* +plot_likelihood_3d(likelihood_3d(Y)) +#+end_src + +#+RESULTS: +[[file:babel_images/density_mu_sigma1.pdf]] + +** TODO Influence of the prior +Take away messages: +1. With enough data, reasonable people *converge*. +2. If any $p(\theta) =0$, no data will change that + - Sometimes imposing $p(\theta)=0$ is nice (e.g., $\theta>0$) +3. An uninformative prior is better than a wrong highly (supposedly) + informative prior. +4. With *conjugate* priors, calculus of the likelihood is + possible + + Otherwise, the normalization is a *huge pain* + +Computing confidence intervals, high density regions, expectation of +complex functions... *Samples* are easier to use than distributions. +#+BEGIN_CENTER + *BUGS*: Bayesian inference Using Gibbs Sampling +#+END_CENTER + + #+BEGIN_CENTER + $\boxed{\underbrace{p(\theta|y,x)}_{\text{\alert{Posterior}}} \propto + \underbrace{p(y|\theta,x)}_{\text{\alert{Likelihood}}}\underbrace{p(\theta,x)}_{\text{\alert{Prior}}}}$ \medskip + #+END_CENTER + + +** COMMENT Possible use for Bayesian Sampling +- Testing: warn on significant difference +- Characterize variability of existing systems +- Multi-armed bandit: Thompson sampling +* COMMENT A Uniform model +** Posterior for a Uniform Model +#+begin_src R :results output :session *R* :exports results +summary(data) +#+end_src + +#+RESULTS: +: Min. 1st Qu. Median Mean 3rd Qu. Max. +: 9.221 11.334 12.180 12.073 12.982 15.447 + +#+ATTR_LATEX: :width 5cm :center nil +[[file:babel_images/hist1.pdf]] +#+ATTR_LATEX: :width 5cm :center nil +# [[file:babel_images/density_alpha_beta1.pdf]] + +*** Graphs :noexport: +** Single point estimate +- Maximum likelihood +- Expectation +* Bayesian Sampling +#+LaTeX: \def\includepic#1{\hfill$\begin{array}{l}\boxed{\includegraphics[width=1.4cm]{babel_images/#1}}\end{array}$}% + +** R code :noexport: + +** Generating random number: direct method +#+begin_src R :results output graphics :file babel_images/direct_gen_runif.pdf :exports none :width 6 :height 4 :session *R* +hist(runif(1000)) +#+end_src + +#+RESULTS: +[[file:babel_images/direct_gen_runif.pdf]] + +#+begin_src R :results output graphics :file babel_images/direct_gen_density.pdf :exports none :width 6 :height 4 :session *R* +plot(dnorm,xlim=c(-5,5)) +#+end_src + +#+RESULTS: +[[file:babel_images/direct_gen_density.pdf]] + +#+begin_src R :results output graphics :file babel_images/direct_gen_pdf.pdf :exports none :width 6 :height 4 :session *R* +plot(pnorm,xlim=c(-5,5)) +#+end_src + +#+RESULTS: +[[file:babel_images/direct_gen_pdf.pdf]] + +#+begin_src R :results output graphics :file babel_images/direct_gen_qf.pdf :exports none :width 6 :height 4 :session *R* +plot(qnorm,xlim=c(0,1)) +#+end_src + +#+RESULTS: +[[file:babel_images/direct_gen_qf.pdf]] + +#+begin_src R :results output graphics :file babel_images/direct_gen_rnorm.pdf :exports none :width 6 :height 4 :session *R* +hist(rnorm(1000)) +#+end_src + +#+RESULTS: +[[file:babel_images/direct_gen_rnorm.pdf]] + +#+begin_src shell :results output :exports none +for i in babel_images/direct_gen_*.pdf ; do + pdfcrop $i $i ; +done +#+end_src + +#+RESULTS: +#+begin_example +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/direct_gen_density.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/direct_gen_pdf.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/direct_gen_qf.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/direct_gen_rnorm.pdf'. +PDFCROP 1.38, 2012/11/02 - Copyright (c) 2002-2012 by Heiko Oberdiek. +==> 1 page written on `babel_images/direct_gen_runif.pdf'. +#+end_example + +# Wait, how does this work btw ? + +- Input:\vspace{-1em} + - $\Unif(0,1)$ \includepic{direct_gen_runif.pdf} + - A target density $f_Y$ \includepic{direct_gen_density.pdf} +- 3 \textit{Easy} steps: + 1. Compute $F_Y(t) = \int_{-\infty}^t f_Y(y).dy$ \includepic{direct_gen_pdf.pdf} + 2. Compute the inverse $F_Y^{-1}$ \includepic{direct_gen_qf.pdf} + 3. Apply $F_Y^{-1}$ to your uniform numbers \includepic{direct_gen_rnorm.pdf} +Step 1 is generally quite complicated. The /prior/ makes it /even worse/. + +Multi-dimensional densities: just as complicated unless the +law has a very particular structure +** Rejection method +#+ATTR_LATEX: :width 5cm +file:img/gelman_264.pdf + +Assume we have $M$ and $g$, s.t. $p(\theta|y)\le Mg(\theta)$ + +- Draw $\theta \sim g$ and accept with probability $\displaystyle \frac{p(\theta|y)}{Mg(\theta)}$ + +Works well if $Mg$ is a good approximation of $p(.|y)$ +*** Issues: +- $p$ is multiplied by the prior. Where is the max? Which $g$, which + $M$? +- Is the landscape flat, hilly, spiky? +- Rejection can be quite inefficient ($\leadsto$ Importance sampling) +** Monte Carlo Markov Chain simulation +Dimension by dimension (*Gibbs sampler*): $\theta_j^t \sim p(.|\theta^{t-1}_{-j},y)$ +#+ATTR_LATEX: :width 7cm +file:img/gelman_277.pdf + +\pause\vspace{-1em} +*Metropolis-Hasting*: Jumping distribution $J$ + - + #+BEGIN_EXPORT latex + $\theta^* \sim J(\theta^{t-1})$\hfill + $\displaystyle r=\frac{p(\theta^*|y)}{p(\theta^{t-1}|y)}$\hfill + $\theta^t = \begin{cases} + \theta^*& \text{with proba. $\min(r,1)$}\\ + \theta^{t-1} & \text{otherwise} + \end{cases}$ + #+END_EXPORT +Look for *high density areas* + - \small Highly skewed (short/long-tail) or multi-modal are problematic + - Transformation, reparameterization, auxiliary variables, simulated + tempering, ... + - *Trans-dimensional Markov chains*: the dimension of the parameter + space can change from one iteration to the next +** Hamiltonian Monte-Carlo +Try to *eliminate the random walk inefficiency* + - Add a momentum variable $\phi_j$ for each component $\theta_j$ and move to + the right direction + +*Hamiltonian Monte-Carlo* combines MCMC with deterministic optimization +methods +- *Leapfrog*: $L$ steps of $\epsilon/2$ ($L\epsilon\approx 1$) +- No U-turn Sampler (*NUTS*): adapt step sizes locally, the trajectory + continues until it turns around +* Using STAN +** What is Stan? +\vspace{1em} +#+LaTeX: \begin{columns} +#+LaTeX: \begin{column}{.3\linewidth} + \includegraphics[width=1.1\linewidth]{img/stanislaw_ulam.png} +#+LaTeX: \end{column} +#+LaTeX: \begin{column}{.73\linewidth}\it + A probabilistic programming language implementing full *Bayesian + statistical inference with MCMC sampling* (NUTS, HMC) and penalized + maximum likelihood estimation with optimization (L-BFGS)\medskip + + \small + *Stanislaw Ulam*, namesake of Stan and co-inventor of Monte Carlo + methods shown here holding the Fermiac, Enrico Fermi’s physical + Monte Carlo simulator for neutron diffusion +#+LaTeX: \end{column} +#+LaTeX: \end{columns}\bigskip + +#+LaTeX: \begin{columns} +#+LaTeX: \begin{column}{.35\linewidth}\centering + \includegraphics[height=3cm]{img/bayes_gelman.jpg} + + \small *Bayesian Data Analysis*, \newline + Gelman et al., 2013 +#+LaTeX: \end{column} +#+LaTeX: \begin{column}{.6\linewidth}\centering + \includegraphics[height=3cm]{img/bayes_mcelreath.jpg} + + \small *Bayesian Course with examples in R and Stan,* \newline + Richard McElreath, 2015 +#+LaTeX: \end{column} +#+LaTeX: \end{columns} +** A simple example + +#+begin_src R :results output :session *R* :exports none +generate_dataset=function(intercept, coefficient, N, min_x=0, max_x=100, sigma=1){ + x = sample(min_x:max_x,N,replace=T) + y = coefficient * x + intercept + rnorm(N,sd=sigma) + df = data.frame(x=x,y=y) + return(df) +} +df=generate_dataset(50, -2, 500, sigma=15) +head(df) +#+end_src + +#+RESULTS: +: x y +: 1 75 -108.999735 +: 2 38 -48.453814 +: 3 3 29.142740 +: 4 52 -54.413985 +: 5 27 -1.098467 +: 6 90 -127.396345 + +#+begin_src R :results output graphics :file babel_images/stan_data.pdf :exports both :width 6 :height 4 :session *R* +ggplot(df, aes(x, y))+geom_point(alpha=0.3) + theme_bw() +#+end_src + +#+RESULTS: +[[file:babel_images/stan_data.pdf]] +** A STAN model +#+begin_example +library(rstan) + +modelString = "data { // the observations + int<lower=1> N; // number of points + vector[N] x; + vector[N] y; +} +parameters { // what we want to find + real intercept; + real coefficient; + real<lower=0> sigma; // indication: sigma cannot be negative +} +model { + // We define our priors + intercept ~ normal(0, 10); // We know that all the parameters follow a normal distribution + coefficient ~ normal(0, 10); + sigma ~ normal(0, 10); + + // Then, our likelihood function + y ~ normal(coefficient*x + intercept, sigma); +} +" +sm = stan_model(model_code = modelString) + +#+end_example +*** True R code :noexport: +#+begin_src R :results output :session *R* :exports code +library(rstan) + +modelString = "data { // the observations + int<lower=1> N; // number of points + vector[N] x; + vector[N] y; +} +parameters { // what we want to find + real intercept; + real coefficient; + real<lower=0> sigma; // indication: sigma cannot be negative +} +model { + // We define our priors + intercept ~ normal(0, 10); // We know that all the parameters follow a normal distribution + coefficient ~ normal(0, 10); + sigma ~ normal(0, 10); + + // Then, our likelihood function + y ~ normal(coefficient*x + intercept, sigma); +} +" +sm = stan_model(model_code = modelString) +#+end_src +** Running STAN +#+begin_src R :results output :session *R* :exports both +data = list(N=nrow(df),x=df$x,y=df$y) +fit = sampling(sm,data=data, iter=500, chains=8) +#+end_src + +#+RESULTS: +#+begin_example +SAMPLING FOR MODEL 'ea4b5a288cf5f1d87215860103a9026e' NOW (CHAIN 1). +Chain 1: Gradient evaluation took 7.6e-05 seconds +Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.76 seconds. +Chain 1: Iteration: 1 / 500 [ 0%] (Warmup) +Chain 1: Iteration: 50 / 500 [ 10%] (Warmup) +Chain 1: Iteration: 100 / 500 [ 20%] (Warmup) +Chain 1: Iteration: 150 / 500 [ 30%] (Warmup) +Chain 1: Iteration: 200 / 500 [ 40%] (Warmup) +Chain 1: Iteration: 250 / 500 [ 50%] (Warmup) +Chain 1: Iteration: 251 / 500 [ 50%] (Sampling) +Chain 1: Iteration: 300 / 500 [ 60%] (Sampling) +Chain 1: Iteration: 350 / 500 [ 70%] (Sampling) +Chain 1: Iteration: 400 / 500 [ 80%] (Sampling) +Chain 1: Iteration: 450 / 500 [ 90%] (Sampling) +Chain 1: Iteration: 500 / 500 [100%] (Sampling) +Chain 1: Elapsed Time: 0.101632 seconds (Warm-up) +Chain 1: 0.044023 seconds (Sampling) +Chain 1: 0.145655 seconds (Total) + +SAMPLING FOR MODEL 'ea4b5a288cf5f1d87215860103a9026e' NOW (CHAIN 2). +Chain 2: Gradient evaluation took 2e-05 seconds +Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds. +Chain 2: Iteration: 1 / 500 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(Sampling) +Chain 8: Elapsed Time: 0.090898 seconds (Warm-up) +Chain 8: 0.048248 seconds (Sampling) +Chain 8: 0.139146 seconds (Total) +Inference for Stan model: ea4b5a288cf5f1d87215860103a9026e. +8 chains, each with iter=500; warmup=250; thin=1; +post-warmup draws per chain=250, total post-warmup draws=2000. + + mean se_mean sd 2.5% 25% 50% 75% 97.5% +intercept 49.12 0.04 1.31 46.53 48.24 49.13 50.00 51.68 +coefficient -1.96 0.00 0.02 -2.01 -1.98 -1.96 -1.95 -1.92 +sigma 15.48 0.01 0.48 14.56 15.15 15.47 15.79 16.44 +lp__ -1630.71 0.04 1.14 -1633.61 -1631.32 -1630.42 -1629.85 -1629.36 + n_eff Rhat +intercept 997 1.00 +coefficient 979 1.00 +sigma 1057 1.00 +lp__ 840 1.01 + +Samples were drawn using NUTS(diag_e) at Wed May 22 22:30:52 2019. +For each parameter, n_eff is a crude measure of effective sample size, +and Rhat is the potential scale reduction factor on split chains (at +convergence, Rhat=1). +#+end_example +** Inspecting results +#+begin_src R :results output :session *R* :exports both +print(fit) +#+end_src + +#+RESULTS: +#+begin_example + mean se_mean sd 2.5% 25% 50% 75% 97.5% +intercept 49.12 0.04 1.31 46.53 48.24 49.13 50.00 51.68 +coefficient -1.96 0.00 0.02 -2.01 -1.98 -1.96 -1.95 -1.92 +sigma 15.48 0.01 0.48 14.56 15.15 15.47 15.79 16.44 +lp__ -1630.71 0.04 1.14 -1633.61 -1631.32 -1630.42 -1629.85 -1629.36 + n_eff Rhat +intercept 997 1.00 +coefficient 979 1.00 +sigma 1057 1.00 +lp__ 840 1.01 + +Samples were drawn using NUTS(diag_e) at Wed May 22 22:30:52 2019. +For each parameter, n_eff is a crude measure of effective sample size, +and Rhat is the potential scale reduction factor on split chains (at +convergence, Rhat=1). +#+end_example +** Checking Convergence +#+begin_src R :results output graphics :file babel_images/stan_convergence.pdf :exports both :width 8 :height 4 :session *R* +stan_trace(fit) +#+end_src + +#+RESULTS: +[[file:babel_images/stan_convergence.pdf]] +** Looking at samples + +#+LaTeX: \begin{columns} +#+LaTeX: \begin{column}[t]{.45\linewidth}\centering + #+begin_src R :results output graphics :file babel_images/stan_hist.pdf :exports both :width 3.5 :height 3.5 :session *R* + stan_hist(fit) + #+end_src + + #+RESULTS: + [[file:babel_images/stan_hist.pdf]] +#+LaTeX: \end{column} +#+LaTeX: \begin{column}[t]{.45\linewidth}\centering + #+begin_src R :results output graphics :file babel_images/stan_samples.pdf :exports both :width 4 :height 4 :session *R* + ggplot(as.data.frame(rstan::extract(fit))) + theme_bw() + + geom_point(aes(x=intercept, y=coefficient, color=sigma)) + #+end_src + + #+RESULTS: + [[file:babel_images/stan_samples.pdf]] +#+LaTeX: \end{column} +#+LaTeX: \end{columns} +* Wrap-up +** Truth vs. Myths + +*** Where it fails: +- +Cover the space+ (e.g., high dimensions) +- +Uninformed far away density spikes+ (mixtures requires informative + models and priors) +- +High quantiles/rare events+ + +Informative priors and starting points may be difficult to come up +with (use *machine learning/simpler estimates* first?) + +*** Where it helps: +- Captures "correlations" +- Robust expectation estimation (1 simulation = very biased) +- Exploit all your knowledge about the system +- Uncertainty quantification with Monte Carlo +** How we would like to use it + +#+BEGIN_CENTER +Disclaimer: I may be *naive* +#+END_CENTER + +- Digital twins :: (haha!) of platforms in *SimGrid* + - As realistic as possible from observations + - Variations (what if?)\medskip + +- Trace analysis: :: expressive models of internal structure + - better information compression/summary + - state what is expected and detect when the system departs from + it\medskip + +- Anomaly detection: :: performance non-regression testing (CI) + - Model the whole system ($y$ in "high" dimension) + - Detect novelty and continuously enrich the model and data + +* Emacs Setup :noexport: +This document has local variables in its postembule, which should +allow Org-mode (9) to work seamlessly without any setup. If you're +uncomfortable using such variables, you can safely ignore them at +startup. 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