diff --git a/ICI3D_Lab_MCMC-Binomial.R b/ICI3D_Lab_MCMC-Binomial.R index 692867b..4b3f466 100644 --- a/ICI3D_Lab_MCMC-Binomial.R +++ b/ICI3D_Lab_MCMC-Binomial.R @@ -1,9 +1,8 @@ -#' Introduction to MCMC 1: Estimating a posterior binomial probability -#' Clinic on the Meaningful Modeling of Epidemiological Data -#' International Clinics on Infectious Disease Dynamics and Data (ICI3D) Program -#' African Institute for Mathematical Sciences, Muizenberg, RSA +#' Introduction to Inference: Understanding MCMC with Binomial Data +#' [MMED](https://www.ici3d.org/MMED) #' Steve Bellan 2012, 2015 -#' Carl Pearson 2025 +#' Carl Pearson 2025, 2026 +#' Claire Perrin Smith 2026 #' #' By the end of this tutorial you should be able to: #' * Write a likelihood function for binomially distributed data @@ -12,6 +11,10 @@ #' * Assess MCMC convergence with the Gelman-Rubin diagnostic #' and trace plots +#' Revisit the notion of population true value => distribution of plausible samples +#' - have them generate the plot, using a functional form they'll use again later? +#' - have them add a particular observation + library(ggplot2) library(dplyr) library(tidyr) @@ -30,6 +33,38 @@ sample_size <- 100 sample_positive <- rbinom(1, sample_size, true_prevalence) sample_prevalence <- sample_positive / sample_size +#' @carl have them make the dbinom link +#' and then insert into our figure code + +#' @carl have them convert that code into a likelihood function +#' and feed that into a plot we provide + +#' @carl have them write a prior "function" to feed to prior + likelihood = posterior plot +#' + +#' @carl build up MCMC answer: +#' - implement a proposal distribution + acceptance rejection logic +#' - insert that into a loop structure that we provide, FROM A STARTING POINT WE PROVIDE +#' - that loop structure should do a pass, draw a plot, prompt for do 1 step more +#' - also provide a run-until-option +#' - MVP++: also provide a save animation option + +#' prompt with "what if we started at a different (bad) place?" (repeat previous, but from +#' different BAD starting point that we provide) + +#' have them run the chain sampling code that we provide - have them provide +#' that method all of their previous inputs, plus the good and bad starting points, +#' and some points of their choice +#' should be another step-step-etc-continue loop, but plot is chains and posteriors +#' needs to show R hat calculation as well + + +#' have them implement the three different proposals, drop them into the chain sampling code +#' and see the final results + +######## EVERYTHING BELOW IS OLD - POTENTIALLY USEFUL FOR ABOVE, BUT NO LONGER NECESSARILY IN CODE + + #' @question How do the true_ and sample_prevalence compare? What could #' you change to make them more likely to be close for any given draw? diff --git a/mcmc_lecture/bayes_rule.pdf b/mcmc_lecture/bayes_rule.pdf new file mode 100644 index 0000000..32099a0 Binary files /dev/null and b/mcmc_lecture/bayes_rule.pdf differ diff --git a/mcmc_lecture/bayes_rule.png b/mcmc_lecture/bayes_rule.png new file mode 100644 index 0000000..9eb2d3b Binary files /dev/null and b/mcmc_lecture/bayes_rule.png differ diff --git a/mcmc_lecture/bayes_rule.tex b/mcmc_lecture/bayes_rule.tex new file mode 100644 index 0000000..d44bb85 --- /dev/null +++ b/mcmc_lecture/bayes_rule.tex @@ -0,0 +1,19 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{P\left(\theta \middle| y\right)} = + \frac{\textcolor{likelihood}{P\left(y \middle| \theta\right)}\textcolor{prior}{P\left(\theta\right)}}{\textcolor{denom}{P\left(y\right)}} + } + \] + +\end{document} + diff --git a/mcmc_lecture/bayes_rule_prop.pdf b/mcmc_lecture/bayes_rule_prop.pdf new file mode 100644 index 0000000..0c66db9 Binary files /dev/null and b/mcmc_lecture/bayes_rule_prop.pdf differ diff --git a/mcmc_lecture/bayes_rule_prop.png b/mcmc_lecture/bayes_rule_prop.png new file mode 100644 index 0000000..57c0994 Binary files /dev/null and b/mcmc_lecture/bayes_rule_prop.png differ diff --git a/mcmc_lecture/bayes_rule_prop.tex b/mcmc_lecture/bayes_rule_prop.tex new file mode 100644 index 0000000..fa25168 --- /dev/null +++ b/mcmc_lecture/bayes_rule_prop.tex @@ -0,0 +1,19 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{P\left(\theta \middle| y\right)} \propto + \textcolor{likelihood}{P\left(y \middle| \theta\right)}\textcolor{prior}{P\left(\theta\right)} + } + \] + +\end{document} + diff --git a/mcmc_lecture/bayes_rule_prop_text.pdf b/mcmc_lecture/bayes_rule_prop_text.pdf new file mode 100644 index 0000000..588285d Binary files /dev/null and b/mcmc_lecture/bayes_rule_prop_text.pdf differ diff --git a/mcmc_lecture/bayes_rule_prop_text.png b/mcmc_lecture/bayes_rule_prop_text.png new file mode 100644 index 0000000..e45947e Binary files /dev/null and b/mcmc_lecture/bayes_rule_prop_text.png differ diff --git a/mcmc_lecture/bayes_rule_prop_text.tex b/mcmc_lecture/bayes_rule_prop_text.tex new file mode 100644 index 0000000..5739802 --- /dev/null +++ b/mcmc_lecture/bayes_rule_prop_text.tex @@ -0,0 +1,19 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{\mathrm{posterior}} \propto + \textcolor{likelihood}{\mathrm{likelihood}}\times\textcolor{prior}{\mathrm{prior}} + } + \] + +\end{document} + diff --git a/mcmc_lecture/bayes_rule_text.pdf b/mcmc_lecture/bayes_rule_text.pdf new file mode 100644 index 0000000..f6f3f77 Binary files /dev/null and b/mcmc_lecture/bayes_rule_text.pdf differ diff --git a/mcmc_lecture/bayes_rule_text.png b/mcmc_lecture/bayes_rule_text.png new file mode 100644 index 0000000..c8c27bc Binary files /dev/null and b/mcmc_lecture/bayes_rule_text.png differ diff --git a/mcmc_lecture/bayes_rule_text.tex b/mcmc_lecture/bayes_rule_text.tex new file mode 100644 index 0000000..132cade --- /dev/null +++ b/mcmc_lecture/bayes_rule_text.tex @@ -0,0 +1,20 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{\mathrm{posterior}} = + \frac{\textcolor{likelihood}{\mathrm{likelihood}}\times\textcolor{prior}{\mathrm{prior}}} + {\int\textcolor{likelihood}{\mathrm{likelihood}}\times\textcolor{prior}{\mathrm{prior}}\ d\theta} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_lik.pdf b/mcmc_lecture/binom_lik.pdf new file mode 100644 index 0000000..c58137f Binary files /dev/null and b/mcmc_lecture/binom_lik.pdf differ diff --git a/mcmc_lecture/binom_lik.png b/mcmc_lecture/binom_lik.png new file mode 100644 index 0000000..e2bdea4 Binary files /dev/null and b/mcmc_lecture/binom_lik.png differ diff --git a/mcmc_lecture/binom_lik.tex b/mcmc_lecture/binom_lik.tex new file mode 100644 index 0000000..117417d --- /dev/null +++ b/mcmc_lecture/binom_lik.tex @@ -0,0 +1,18 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{likelihood}{P\left(x=28,\ n=100\middle|\theta\right)} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_lik_ex.pdf b/mcmc_lecture/binom_lik_ex.pdf new file mode 100644 index 0000000..85985fd Binary files /dev/null and b/mcmc_lecture/binom_lik_ex.pdf differ diff --git a/mcmc_lecture/binom_lik_ex.png b/mcmc_lecture/binom_lik_ex.png new file mode 100644 index 0000000..e3983d6 Binary files /dev/null and b/mcmc_lecture/binom_lik_ex.png differ diff --git a/mcmc_lecture/binom_lik_ex.tex b/mcmc_lecture/binom_lik_ex.tex new file mode 100644 index 0000000..117417d --- /dev/null +++ b/mcmc_lecture/binom_lik_ex.tex @@ -0,0 +1,18 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{likelihood}{P\left(x=28,\ n=100\middle|\theta\right)} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_mcmc_ex.pdf b/mcmc_lecture/binom_mcmc_ex.pdf new file mode 100644 index 0000000..a44e9c2 Binary files /dev/null and b/mcmc_lecture/binom_mcmc_ex.pdf differ diff --git a/mcmc_lecture/binom_mcmc_ex.png b/mcmc_lecture/binom_mcmc_ex.png new file mode 100644 index 0000000..3b85ba2 Binary files /dev/null and b/mcmc_lecture/binom_mcmc_ex.png differ diff --git a/mcmc_lecture/binom_mcmc_ex.tex b/mcmc_lecture/binom_mcmc_ex.tex new file mode 100644 index 0000000..0d97846 --- /dev/null +++ b/mcmc_lecture/binom_mcmc_ex.tex @@ -0,0 +1,20 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{P\left(\theta\middle|x=28,\ n=100\right)}\ \propto\ + \textcolor{likelihood}{P\left(x=28,\ n=100\middle|\theta\right)} + \textcolor{prior}{P\left(\theta\right)} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_mcmc_full_ex.pdf b/mcmc_lecture/binom_mcmc_full_ex.pdf new file mode 100644 index 0000000..6893f05 Binary files /dev/null and b/mcmc_lecture/binom_mcmc_full_ex.pdf differ diff --git a/mcmc_lecture/binom_mcmc_full_ex.png b/mcmc_lecture/binom_mcmc_full_ex.png new file mode 100644 index 0000000..f8a60ee Binary files /dev/null and b/mcmc_lecture/binom_mcmc_full_ex.png differ diff --git a/mcmc_lecture/binom_mcmc_full_ex.tex b/mcmc_lecture/binom_mcmc_full_ex.tex new file mode 100644 index 0000000..6d5f9d7 --- /dev/null +++ b/mcmc_lecture/binom_mcmc_full_ex.tex @@ -0,0 +1,20 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{P\left(\theta\middle|x=28,\ n=100\right)}= + \frac{\textcolor{likelihood}{P\left(x=28,\ n=100\middle|\theta\right)}\textcolor{prior}{P\left(\theta\right)}} + {\int\textcolor{likelihood}{P\left(x=28,\ n=100\middle|\theta\right)}\textcolor{prior}{P\left(\theta\right)}\ d\theta} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_pdf.pdf b/mcmc_lecture/binom_pdf.pdf new file mode 100644 index 0000000..f3ab962 Binary files /dev/null and b/mcmc_lecture/binom_pdf.pdf differ diff --git a/mcmc_lecture/binom_pdf.png b/mcmc_lecture/binom_pdf.png new file mode 100644 index 0000000..6f296c7 Binary files /dev/null and b/mcmc_lecture/binom_pdf.png differ diff --git a/mcmc_lecture/binom_pdf.tex b/mcmc_lecture/binom_pdf.tex new file mode 100644 index 0000000..6ec8612 --- /dev/null +++ b/mcmc_lecture/binom_pdf.tex @@ -0,0 +1,18 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + f\left(x\middle|p\right)={n\choose x} \left(p\right)^x \left(1-p\right)^{n-x} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_posterior_ex.pdf b/mcmc_lecture/binom_posterior_ex.pdf new file mode 100644 index 0000000..e8e6d79 Binary files /dev/null and b/mcmc_lecture/binom_posterior_ex.pdf differ diff --git a/mcmc_lecture/binom_posterior_ex.png b/mcmc_lecture/binom_posterior_ex.png new file mode 100644 index 0000000..b1a1054 Binary files /dev/null and b/mcmc_lecture/binom_posterior_ex.png differ diff --git a/mcmc_lecture/binom_posterior_ex.tex b/mcmc_lecture/binom_posterior_ex.tex new file mode 100644 index 0000000..b3d319f --- /dev/null +++ b/mcmc_lecture/binom_posterior_ex.tex @@ -0,0 +1,18 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \textcolor{posterior}{P\left(\theta\middle|x=28,\ n=100\right)} + } + \] + +\end{document} + diff --git a/mcmc_lecture/binom_posterior_ex_integrate.pdf b/mcmc_lecture/binom_posterior_ex_integrate.pdf new file mode 100644 index 0000000..37cc193 Binary files /dev/null and b/mcmc_lecture/binom_posterior_ex_integrate.pdf differ diff --git a/mcmc_lecture/binom_posterior_ex_integrate.png b/mcmc_lecture/binom_posterior_ex_integrate.png new file mode 100644 index 0000000..0242690 Binary files /dev/null and b/mcmc_lecture/binom_posterior_ex_integrate.png differ diff --git a/mcmc_lecture/binom_posterior_ex_integrate.tex b/mcmc_lecture/binom_posterior_ex_integrate.tex new file mode 100644 index 0000000..0f03585 --- /dev/null +++ b/mcmc_lecture/binom_posterior_ex_integrate.tex @@ -0,0 +1,18 @@ + +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \int\textcolor{posterior}{P\left(\theta\middle|x=28,\ n=100\right)}\ d\theta=1 + } + \] + +\end{document} + diff --git a/mcmc_lecture/colors.aux b/mcmc_lecture/colors.aux new file mode 100644 index 0000000..e54b19f --- /dev/null +++ b/mcmc_lecture/colors.aux @@ -0,0 +1,21 @@ +\relax +\@setckpt{colors}{ +\setcounter{page}{1} +\setcounter{equation}{0} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{sapage}{0} +\setcounter{part}{0} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{figure}{0} +\setcounter{table}{0} +\setcounter{parentequation}{0} +} diff --git a/mcmc_lecture/colors.tex b/mcmc_lecture/colors.tex new file mode 100644 index 0000000..9b10eb8 --- /dev/null +++ b/mcmc_lecture/colors.tex @@ -0,0 +1,8 @@ +\usepackage{xcolor} + +% Define your colors +\definecolor{posterior}{HTML}{00b300} % green +\definecolor{likelihood}{HTML}{cc00ff} % purple +\definecolor{prior}{HTML}{ff9933} % orange +\definecolor{denom}{HTML}{0099ff} % Med blue +\definecolor{default}{HTML}{000000} % Black \ No newline at end of file diff --git a/mcmc_lecture/pdf_integrate_to_1.pdf b/mcmc_lecture/pdf_integrate_to_1.pdf new file mode 100644 index 0000000..4719efb Binary files /dev/null and b/mcmc_lecture/pdf_integrate_to_1.pdf differ diff --git a/mcmc_lecture/pdf_integrate_to_1.png b/mcmc_lecture/pdf_integrate_to_1.png new file mode 100644 index 0000000..eb68664 Binary files /dev/null and b/mcmc_lecture/pdf_integrate_to_1.png differ diff --git a/mcmc_lecture/pdf_integrate_to_1.tex b/mcmc_lecture/pdf_integrate_to_1.tex 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Missing $ inserted. + + $ +l.1 + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.1 + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \tex_par:D +l.1 + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Missing $ inserted. + + $ +l.2 ...posterior}{P\left(\theta \middle| y\right)} + = +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\@textcolor ...otect \leavevmode {\color #1{#2}#3} + +l.2 ...posterior}{P\left(\theta \middle| y\right)} + = +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +) +! Extra }, or forgotten $. +\@textcolor ...otect \leavevmode {\color #1{#2}#3} + +l.13 } + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.14 \] + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. + + } +l.14 \] + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +[1 + + +Non-PDF special ignored! + papersize=84.89398pt,37.19841pt +{C:/Users/cpsmit/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map}] +(wrapper.aux) + *********** +LaTeX2e <2025-11-01> +L3 programming layer <2026-04-28> + *********** + ) +Here is how much of TeX's memory you used: + 3809 strings out of 467683 + 62434 string characters out of 5414237 + 476033 words of memory out of 5000000 + 32845 multiletter control sequences out of 15000+600000 + 629420 words of font info for 48 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 70i,7n,79p,263b,199s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on wrapper.pdf (1 page, 54124 bytes). +PDF statistics: + 36 PDF objects out of 1000 (max. 8388607) + 0 named destinations out of 1000 (max. 500000) + 1 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/mcmc_lecture/wrapper.pdf b/mcmc_lecture/wrapper.pdf new file mode 100644 index 0000000..1515f3f Binary files /dev/null and b/mcmc_lecture/wrapper.pdf differ diff --git a/mcmc_lecture/wrapper.synctex.gz b/mcmc_lecture/wrapper.synctex.gz new file mode 100644 index 0000000..707cb3b Binary files /dev/null and b/mcmc_lecture/wrapper.synctex.gz differ diff --git a/mcmc_lecture/wrapper.tex b/mcmc_lecture/wrapper.tex new file mode 100644 index 0000000..d3d0848 --- /dev/null +++ b/mcmc_lecture/wrapper.tex @@ -0,0 +1,16 @@ +\documentclass[varwidth,border=1pt]{standalone} + +\input{colors.tex} + +\usepackage{mathtools} +\usepackage{amsfonts} + +\begin{document} + + \[ + \textcolor{default}{ + \input{} + } + \] + +\end{document} \ No newline at end of file