Bayesian Statistics Course

Lesson 1 • Joint, Marginal, and Conditional Distributions

Examines multivariate probability structures essential for Bayesian updating. Connects joint distributions to the mechanics of Bayes' theorem.

Lesson 2 • Random Variables and Distributions

Introduces discrete and continuous random variables and their distributions. Provides the distributional vocabulary needed for Bayesian modelling.

Lesson 3 • Frequentist vs. Bayesian Perspectives

Contrasts the two major schools of statistical inference. Students identify where each approach is appropriate and why Bayesian methods offer unique advantages.

Lesson 1 • Prior, Likelihood, and Posterior

Defines the three core components of Bayesian inference and their relationships. Students interpret each component's role in updating beliefs with data.

Lesson 2 • Deriving Bayes' Theorem

Derives Bayes' theorem from the definition of conditional probability. Grounds the formula in the probability axioms established in Chapter 1.

Lesson 3 • Simple Worked Examples

Applies Bayes' theorem to classic discrete problems such as medical testing and spam filtering. Builds computational intuition before moving to continuous cases.

Lesson 1 • Likelihood Functions for Continuous Data

Derives Gaussian, Exponential, and Beta likelihoods for continuous outcomes. Prepares students to model real-valued measurements in Bayesian frameworks.

Lesson 2 • Bayesian Linear Regression Model

Builds a complete Bayesian linear regression model with priors on coefficients and noise. Serves as the primary applied example for subsequent computational chapters.

Lesson 3 • Generative Model Specification

Introduces the generative modelling perspective where data are treated as draws from a parameterised process. Students write full probabilistic model descriptions.

Lesson 1 • Conjugate Prior Families

Introduces conjugate priors that yield analytically tractable posteriors. Students match conjugate families to common likelihood models.

Lesson 2 • Eliciting Priors from Domain Experts

Covers structured techniques for translating expert knowledge into probability distributions. Connects prior elicitation to reproducible and defensible modelling practice.

Lesson 3 • Prior Sensitivity Analysis

Teaches systematic comparison of posteriors under alternative prior specifications. Students quantify how much conclusions depend on prior assumptions.

Lesson 1 • Posterior Predictive Distributions

Derives predictive distributions by marginalising over posterior uncertainty. Students use predictive distributions for forecasting and model checking.

Lesson 2 • Conjugate Analysis in Practice

Applies conjugate prior-likelihood pairs to derive closed-form posteriors. Students practise the full update cycle from prior to posterior to predictive.

Lesson 3 • Limitations of Analytical Methods

Identifies when conjugacy fails and analytical solutions become intractable. Motivates the numerical and sampling methods introduced in the next chapter.

Lesson 1 • MCMC Diagnostics and Convergence

Covers trace plots, R-hat, effective sample size, and autocorrelation function as convergence diagnostics. Students identify and remediate poorly mixing chains.

Lesson 2 • Hamiltonian Monte Carlo

Introduces HMC as a gradient-based sampler that reduces random-walk behaviour. Students understand why HMC is preferred for high-dimensional posteriors.

Lesson 3 • Monte Carlo Integration Basics

Introduces Monte Carlo estimation as a general strategy for computing integrals via sampling. Establishes the theoretical basis for MCMC methods.

Lesson 4 • Gibbs Sampling

Presents Gibbs sampling as a special case of Metropolis-Hastings using full conditional distributions. Students implement Gibbs samplers for hierarchical models.

Lesson 1 • Bayes Factors

Introduces Bayes factors as ratios of model evidences for pairwise model comparison. Students interpret Bayes factor scales and apply them to hypothesis testing.

Lesson 2 • Bayesian Model Evidence

Defines marginal likelihood as the Bayesian measure of model fit integrated over parameters. Students compute and interpret model evidence for simple models.

Lesson 3 • Information Criteria for Bayesian Models

Covers WAIC and LOO-CV as practical model comparison tools that avoid marginal likelihood computation. Students apply these criteria using posterior samples.

Lesson 1 • Fitting Hierarchical Models with MCMC

Applies Gibbs and HMC samplers to hierarchical models and addresses funnel geometry challenges. Students use reparameterisation to improve sampler efficiency.

Lesson 2 • Hierarchical Regression Models

Extends hierarchical structure to regression settings with varying intercepts and slopes. Students fit and interpret mixed-effects Bayesian regression models.

Lesson 3 • Advanced Hierarchical Structures

Introduces cross-classified and three-level hierarchical models for complex data structures. Students extend two-level intuition to deeper and non-nested hierarchies.

Lesson 4 • Two-Level Hierarchical Model Structure

Specifies a two-level model with group-level and population-level parameters. Students write the full generative model and identify hyperprior choices.