1  Introduction

The posterior distribution is obtained from the prior distribution and sampling model via Bayes’ rule:

\[p(\theta \mid y)=\frac{p(y \mid \theta) p(\theta)}{\int_{\Theta} p(y \mid \tilde{\theta}) p(\tilde{\theta}) d \tilde{\theta}}.\]

This is a book created from markdown and executable code.

See Knuth (1984) for additional discussion of literate programming.

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1.1 Why Bayesian?

2 Course Topics and Schedule

Week Topics Key Concepts / Readings Computing Focus
1 Introduction to Bayesian Thinking Bayesian vs. Frequentist paradigms; Prior, likelihood, posterior Review of R basics and reproducible workflows
2 Bayesian Inference for Simple Models Conjugate priors, Beta-Binomial, Normal-Normal, Poisson-Gamma Simulating posteriors, visualization
3 Prior Elicitation and Sensitivity Informative vs. noninformative priors, Jeffreys prior Prior sensitivity plots
4 Monte Carlo Integration Law of large numbers, sampling-based inference Random sampling and Monte Carlo approximation
5 Markov Chain Monte Carlo (MCMC) Metropolis-Hastings, Gibbs sampler Implementing MCMC in R
6 Convergence Diagnostics Trace plots, autocorrelation, Gelman–Rubin statistic coda, rstan, and bayesplot packages
7 Hierarchical Bayesian Models Partial pooling, shrinkage, multilevel structures rstanarm / brms
8 Midterm Project: Bayesian Linear Regression Posterior inference for regression, model selection brms, rstanarm, custom Gibbs samplers
9 Bayesian Model Comparison Bayes factors, BIC, DIC, WAIC, LOO Practical comparison via cross-validation
10 Model Checking and Diagnostics Posterior predictive checks, residual analysis pp_check in brms
11 Advanced Computation Hamiltonian Monte Carlo (HMC), Variational Inference Using Stan and CmdStanR
12 Bayesian Decision Theory Utility functions, decision rules, loss minimization Simple decision problems in R
13 Modern Bayesian Methods Approximate Bayesian computation (ABC), Bayesian neural networks Examples via rstan or tensorflow-probability
14 Student Project Presentations Applications and case studies Full workflow demonstration in R

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