Add GPT-5.4 Codex experiment config and generated deliverables

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deliverable/bayesian_inference_beamer.tex

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+\documentclass[aspectratio=169]{beamer}
+\usetheme{Madrid}
+\usecolortheme{default}
+
+\usepackage{amsmath,amssymb}
+\usepackage{bm}
+\usepackage{booktabs}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit,shapes.geometric}
+
+\title{Bayesian Inference}
+\subtitle{A concise introduction for graduate students in quantitative fields}
+\author{Generated self-contained Beamer presentation}
+\date{\today}
+
+\newcommand{\E}{\mathbb{E}}
+\newcommand{\Var}{\mathrm{Var}}
+\newcommand{\Prb}{\mathbb{P}}
+\newcommand{\Normal}{\mathcal{N}}
+\newcommand{\BetaD}{\mathrm{Beta}}
+\newcommand{\Binom}{\mathrm{Binomial}}
+\newcommand{\Bern}{\mathrm{Bernoulli}}
+\newcommand{\iid}{\stackrel{\mathrm{iid}}{\sim}}
+
+\begin{document}
+
+\begin{frame}
+  \titlepage
+\end{frame}
+
+\begin{frame}{Learning goals}
+\begin{itemize}
+  \item Interpret Bayesian inference as coherent updating of uncertainty.
+  \item Distinguish the roles of the prior, likelihood, posterior, and predictive distribution.
+  \item Work through two conjugate examples: Beta--Binomial and Normal--Normal.
+  \item Understand how Bayesian intervals, prediction, computation, and model checking differ from frequentist analogs.
+\end{itemize}
+\vspace{0.5em}
+\begin{block}{Core idea}
+Unknown quantities are treated as random variables, and observed data update beliefs through Bayes' rule.
+\end{block}
+\end{frame}
+
+\begin{frame}{Why Bayesian inference?}
+\begin{columns}[T,onlytextwidth]
+\column{0.57\textwidth}
+\begin{itemize}
+  \item Quantifies uncertainty directly about parameters of interest.
+  \item Naturally combines prior knowledge with new evidence.
+  \item Produces full distributions, not just point estimates.
+  \item Handles prediction and sequential learning in a unified framework.
+\end{itemize}
+
+\column{0.4\textwidth}
+\centering
+\begin{tikzpicture}[>=Latex, node distance=1.3cm]
+  \tikzstyle{box}=[draw, rounded corners, align=center, minimum width=2.8cm, minimum height=0.9cm, fill=blue!8]
+  \node[box] (prior) {Prior belief\\$p(\theta)$};
+  \node[box, below=of prior, fill=green!8] (data) {Observed data\\$y$};
+  \node[box, right=1.0cm of $(prior)!0.5!(data)$, fill=orange!12] (post) {Updated belief\\$p(\theta\mid y)$};
+  \draw[->, thick] (prior.east) -- ++(0.45,0) |- (post.west);
+  \draw[->, thick] (data.east) -- ++(0.45,0) |- (post.west);
+\end{tikzpicture}
+\end{columns}
+\end{frame}
+
+\begin{frame}{Bayes' theorem}
+\begin{block}{Posterior distribution}
+\[
+  p(\theta\mid y)=\frac{p(y\mid\theta)\,p(\theta)}{p(y)},
+  \qquad
+  p(y)=\int p(y\mid\theta)p(\theta)\,d\theta.
+\]
+\end{block}
+
+\begin{columns}[T,onlytextwidth]
+\column{0.55\textwidth}
+\begin{itemize}
+  \item $p(\theta)$: \alert{prior} encodes beliefs before data.
+  \item $p(y\mid\theta)$: \alert{likelihood} measures compatibility of $\theta$ with the data.
+  \item $p(y)$: \alert{evidence} normalizes the posterior.
+  \item $p(\theta\mid y)$: \alert{posterior} combines prior information and data.
+\end{itemize}
+
+\column{0.42\textwidth}
+\begin{block}{Working proportionality}
+In practice we often use
+\[
+  p(\theta\mid y)\propto p(y\mid\theta)p(\theta),
+\]
+and ignore $p(y)$ until normalization or sampling.
+\end{block}
+\end{columns}
+\end{frame}
+
+\begin{frame}{The Bayesian workflow}
+\centering
+\begin{tikzpicture}[>=Latex, node distance=1.35cm]
+  \tikzstyle{stage}=[draw, rounded corners, align=center, minimum width=2.2cm, minimum height=1.0cm, fill=blue!7]
+  \node[stage] (model) {Specify\\model};
+  \node[stage, right=of model] (prior) {Choose\\prior};
+  \node[stage, right=of prior] (update) {Update with\\data};
+  \node[stage, right=of update] (summ) {Summarize\\posterior};
+  \node[stage, below=1.3cm of update, fill=green!10] (check) {Check fit \\\& predict};
+
+  \draw[->, thick] (model) -- (prior);
+  \draw[->, thick] (prior) -- (update);
+  \draw[->, thick] (update) -- (summ);
+  \draw[->, thick] (summ) |- (check);
+  \draw[->, thick] (check.west) -| (model.south);
+\end{tikzpicture}
+
+\vspace{0.8em}
+\begin{itemize}
+  \item Bayesian analysis is iterative: model building and model checking form a loop.
+  \item Posterior predictive checks often reveal misfit even when parameter estimates look reasonable.
+\end{itemize}
+\end{frame}
+
+\begin{frame}{Example 1: Beta--Binomial model}
+Suppose $y$ successes are observed in $n$ Bernoulli trials with success probability $\theta$.
+\[
+  y\mid\theta \sim \Binom(n,\theta),
+  \qquad
+  p(y\mid\theta)\propto \theta^y(1-\theta)^{n-y}.
+\]
+Choose a Beta prior:
+\[
+  \theta \sim \BetaD(\alpha,\beta),
+  \qquad
+  p(\theta)\propto \theta^{\alpha-1}(1-\theta)^{\beta-1}.
+\]
+
+\begin{block}{Conjugacy}
+The posterior is in the same family as the prior, which makes updating algebraically simple.
+\end{block}
+\end{frame}
+
+\begin{frame}{Closed-form update and interpretation}
+Combining likelihood and prior gives
+\[
+  p(\theta\mid y)\propto \theta^{y+\alpha-1}(1-\theta)^{n-y+\beta-1},
+\]
+so that
+\[
+  \theta\mid y \sim \BetaD(\alpha+y,\beta+n-y).
+\]
+
+\begin{columns}[T,onlytextwidth]
+\column{0.52\textwidth}
+\begin{block}{Posterior mean}
+\[
+  \E[\theta\mid y]=\frac{\alpha+y}{\alpha+\beta+n}.
+\]
+It is a weighted average of the prior mean $\alpha/(\alpha+\beta)$ and the sample proportion $y/n$.
+\end{block}
+
+\column{0.44\textwidth}
+\centering
+\vspace{-0.9em}
+\begin{tikzpicture}[>=Latex, node distance=0.58cm, scale=0.8, transform shape]
+  \tikzstyle{mini}=[draw, rounded corners, align=center, minimum width=2.45cm, minimum height=0.62cm]
+  \node[mini, fill=blue!8] (prior) {Prior pseudo-counts\\$\alpha-1,\ \beta-1$};
+  \node[mini, below=of prior, fill=green!8] (data) {Observed counts\\$y,\ n-y$};
+  \node[mini, below=of data, fill=orange!12] (post) {Posterior counts\\$\alpha+y,\ \beta+n-y$};
+  \draw[->, thick] (prior) -- (data);
+  \draw[->, thick] (data) -- (post);
+\end{tikzpicture}
+\end{columns}
+\end{frame}
+
+\begin{frame}{Numerical update example}
+Assume a prior centered at $0.50$ with moderate strength:
+\[
+  \theta \sim \BetaD(4,4).
+\]
+Observe $y=16$ successes in $n=20$ trials. Then
+\[
+  \theta\mid y \sim \BetaD(20,8).
+\]
+
+\begin{columns}[T,onlytextwidth]
+\column{0.55\textwidth}
+\begin{itemize}
+  \item Prior mean: $4/(4+4)=0.50$.
+  \item Sample proportion: $16/20=0.80$.
+  \item Posterior mean: $20/28\approx 0.714$.
+  \item The posterior shrinks the raw sample proportion toward the prior mean.
+\end{itemize}
+
+\column{0.4\textwidth}
+\begin{block}{Interpretation}
+The prior acts like extra observations, so Bayesian estimates often stabilize noisy small-sample problems.
+\end{block}
+\end{columns}
+\end{frame}
+
+\begin{frame}{Example 2: Normal mean with known variance}
+Suppose
+\[
+  y_i \mid \theta \iid \Normal(\theta,\sigma^2),
+  \qquad
+  \theta \sim \Normal(\mu_0,\tau_0^2),
+\]
+with known sampling variance $\sigma^2$.
+
+\begin{block}{Posterior distribution}
+\[
+  \theta\mid y \sim \Normal(\mu_n,\tau_n^2),
+\]
+where
+\[
+  \tau_n^2=\left(\frac{1}{\tau_0^2}+\frac{n}{\sigma^2}\right)^{-1},
+  \qquad
+  \mu_n=\tau_n^2\left(\frac{\mu_0}{\tau_0^2}+\frac{n\bar y}{\sigma^2}\right).
+\]
+\end{block}
+The posterior mean is a precision-weighted average of the prior mean and sample mean.
+\end{frame}
+
+\begin{frame}{Posterior summaries and interval estimates}
+Once we have $p(\theta\mid y)$, common summaries include:
+\[
+  \text{posterior mean }\E[\theta\mid y],
+  \qquad
+  \text{MAP }\arg\max_{\theta} p(\theta\mid y),
+  \qquad
+  \text{posterior variance }\Var(\theta\mid y).
+\]
+
+\begin{columns}[T,onlytextwidth]
+\column{0.5\textwidth}
+\begin{block}{Credible interval}
+A $95\%$ credible interval $[a,b]$ satisfies
+\[
+  \Prb(\theta\in[a,b]\mid y)=0.95.
+\]
+This is a probability statement about the parameter given the observed data.
+\end{block}
+
+\column{0.47\textwidth}
+\begin{block}{Frequentist confidence interval}
+A $95\%$ confidence interval is a procedure whose long-run coverage is $95\%$ over repeated samples.
+\end{block}
+\end{columns}
+\end{frame}
+
+\begin{frame}{Posterior predictive distribution}
+Prediction integrates over parameter uncertainty:
+\[
+  p(\tilde y\mid y)=\int p(\tilde y\mid\theta)p(\theta\mid y)\,d\theta.
+\]
+
+\begin{columns}[T,onlytextwidth]
+\column{0.56\textwidth}
+\begin{itemize}
+  \item This is crucial: predictions should reflect uncertainty in both noise and parameters.
+  \item In the Beta--Binomial model, the predictive probability of success on the next trial is
+  \[
+    \Prb(\tilde y=1\mid y)=\E[\theta\mid y]=\frac{\alpha+y}{\alpha+\beta+n}.
+  \]
+  \item Posterior predictive checks compare replicated data $\tilde y$ with observed data $y$.
+\end{itemize}
+
+\column{0.38\textwidth}
+\centering
+\begin{tikzpicture}[>=Latex, node distance=1.0cm]
+  \tikzstyle{box}=[draw, rounded corners, align=center, minimum width=2.6cm, minimum height=0.85cm]
+  \node[box, fill=orange!12] (post) {Posterior\\$p(\theta\mid y)$};
+  \node[box, below=of post, fill=green!10] (pred) {Predict new data\\$p(\tilde y\mid y)$};
+  \draw[->, thick] (post) -- (pred);
+\end{tikzpicture}
+\end{columns}
+\end{frame}
+
+\begin{frame}{Hierarchical models and partial pooling}
+Bayesian models scale naturally to grouped data. For groups $j=1,\dots,J$:
+\[
+  y_{ij}\mid\theta_j \sim p(y_{ij}\mid\theta_j),
+  \qquad
+  \theta_j\mid\mu,\tau^2 \sim \Normal(\mu,\tau^2).
+\]
+
+\begin{columns}[T,onlytextwidth]
+\column{0.52\textwidth}
+\begin{itemize}
+  \item Group-specific parameters borrow strength from one another.
+  \item Small groups are shrunk more strongly toward the population mean.
+  \item This often improves estimation and prediction relative to no pooling or complete pooling.
+\end{itemize}
+
+\column{0.42\textwidth}
+\centering
+\begin{tikzpicture}[>=Latex]
+  \node[draw, circle, fill=blue!8] (mu) at (0,1.8) {$\mu,\tau^2$};
+  \node[draw, circle, fill=green!10] (t1) at (-1.4,0.5) {$\theta_1$};
+  \node[draw, circle, fill=green!10] (t2) at (0,0.5) {$\theta_2$};
+  \node[draw, circle, fill=green!10] (t3) at (1.4,0.5) {$\theta_J$};
+  \node[draw, circle, fill=orange!10] (y1) at (-1.4,-0.8) {$y_{i1}$};
+  \node[draw, circle, fill=orange!10] (y2) at (0,-0.8) {$y_{i2}$};
+  \node[draw, circle, fill=orange!10] (y3) at (1.4,-0.8) {$y_{iJ}$};
+  \draw[->, thick] (mu) -- (t1);
+  \draw[->, thick] (mu) -- (t2);
+  \draw[->, thick] (mu) -- (t3);
+  \draw[->, thick] (t1) -- (y1);
+  \draw[->, thick] (t2) -- (y2);
+  \draw[->, thick] (t3) -- (y3);
+  \draw[rounded corners] (-2.1,-1.35) rectangle (2.1,0.95);
+  \node at (1.75,-1.15) {$j=1,\dots,J$};
+\end{tikzpicture}
+\end{columns}
+\end{frame}
+
+\begin{frame}{When closed forms fail: computation}
+Many useful posteriors are not analytically tractable.
+
+\begin{block}{Common computational strategies}
+\begin{itemize}
+  \item \textbf{MCMC}: constructs a Markov chain whose stationary distribution is the posterior.
+  \item \textbf{Hamiltonian Monte Carlo}: efficient for high-dimensional continuous parameters.
+  \item \textbf{Variational inference}: turns inference into optimization for faster approximations.
+\end{itemize}
+\end{block}
+
+\begin{block}{Diagnostics matter}
+Check convergence, effective sample size, Monte Carlo standard errors, and sensitivity to priors.
+\end{block}
+\end{frame}
+
+\begin{frame}{Model checking and sensitivity analysis}
+\begin{columns}[T,onlytextwidth]
+\column{0.58\textwidth}
+\begin{itemize}
+  \item \textbf{Posterior predictive checks}: compare observed summaries $T(y)$ to replicated summaries $T(\tilde y)$.
+  \item \textbf{Residual structure}: look for patterns left unexplained by the model.
+  \item \textbf{Prior sensitivity}: ask whether substantive conclusions change under reasonable alternative priors.
+  \item \textbf{Decision relevance}: assess whether posterior uncertainty is small enough for the scientific or policy question.
+\end{itemize}
+
+\column{0.37\textwidth}
+\centering
+\begin{tikzpicture}[>=Latex, node distance=0.95cm]
+  \tikzstyle{box}=[draw, rounded corners, align=center, minimum width=2.6cm, minimum height=0.85cm]
+  \node[box, fill=blue!8] (fit) {Fit model};
+  \node[box, below=of fit, fill=green!8] (rep) {Simulate\\replicated data};
+  \node[box, below=of rep, fill=orange!12] (cmp) {Compare\\$y$ and $\tilde y$};
+  \node[box, below=of cmp, fill=red!10] (rev) {Revise if needed};
+  \draw[->, thick] (fit) -- (rep);
+  \draw[->, thick] (rep) -- (cmp);
+  \draw[->, thick] (cmp) -- (rev);
+  \draw[->, thick] (rev.west) -| ++(-1.0,0) |- (fit.west);
+\end{tikzpicture}
+\end{columns}
+\end{frame}
+
+\begin{frame}{Bayesian linear regression in one line}
+For the Gaussian linear model
+\[
+  \bm y\mid\bm\beta,\sigma^2 \sim \Normal(X\bm\beta,\sigma^2 I),
+\]
+with Gaussian prior $\bm\beta\sim\Normal(\bm\beta_0,V_0)$, the posterior is also Gaussian:
+\[
+  V_n=(V_0^{-1}+X^TX/\sigma^2)^{-1},
+  \qquad
+  \bm\beta_n=V_n\left(V_0^{-1}\bm\beta_0+X^T\bm y/\sigma^2\right).
+\]
+
+\begin{block}{Why this matters}
+This reveals a general pattern: regularization methods such as ridge regression can often be interpreted as Bayesian estimation with specific priors.
+\end{block}
+\end{frame}
+
+\begin{frame}{Takeaways}
+\begin{enumerate}
+  \item Bayesian inference updates uncertainty via
+  \[
+    \text{posterior} \propto \text{likelihood} \times \text{prior}.
+  \]
+  \item Conjugate models build intuition; modern computation handles richer models.
+  \item Credible intervals and predictive distributions are direct, interpretable posterior summaries.
+  \item Good Bayesian practice includes prior choice, computation, model checking, and sensitivity analysis.
+\end{enumerate}
+
+\vspace{0.5em}
+\begin{block}{Final message}
+Bayesian inference is not only a formula; it is a workflow for learning from data under uncertainty.
+\end{block}
+\end{frame}
+
+\end{document}