MCMC Basics

Summary

Chapter 11 of BDA3 introduces Markov chain Monte Carlo — the workhorse of Bayesian computation. MCMC constructs a Markov chain whose stationary distribution is the posterior, enabling sampling from complex, high-dimensional posteriors.

Gibbs Sampler

Iteratively sample each parameter from its full conditional distribution:

$${\theta }_j^{(t+1)}\sim p({\theta }_j\mid {\theta }_{-j}^{(t)},y)$$

• Requires known conditional distributions (often available for conjugate models)
• Each step updates one parameter block, cycling through all blocks
• Can be slow when parameters are highly correlated

Metropolis-Hastings Algorithm

More general: propose a move ${\theta }^{\ast }\sim J({\theta }^{\ast }\mid {\theta }^{(t)})$ and accept with probability:

$$\min \left(1,\frac{p({\theta }^{\ast }\mid y)J({\theta }^{(t)}\mid {\theta }^{\ast })}{p({\theta }^{(t)}\mid y)J({\theta }^{\ast }\mid {\theta }^{(t)})}\right)$$

• Random walk Metropolis: $J({\theta }^{\ast }\mid \theta )=N(\theta ,c^2\Sigma )$ — simple but can be slow in high dimensions
• Optimal acceptance rate: ~0.44 in 1D, ~0.23 in high dimensions

Convergence Diagnostics

• $\hat{R}$ statistic: compare between-chain and within-chain variance across $m$ parallel chains. $\hat{R}<1.1$ indicates approximate convergence
• Effective sample size $n_{\text{eff}}$: accounts for autocorrelation in the chain
• Run multiple chains from dispersed starting points
• Discard warmup/burn-in iterations

See Also

• Introduction to Bayesian Computation — simpler methods for easier problems
• Efficient MCMC — HMC and NUTS, the modern standard
• Computational Troubleshooting — when MCMC goes wrong
• Fitting and Validating Computation — workflow context: how long to run, fake-data checks
• Bayesian Workflow - Overview — MCMC as one step in the full iterative cycle
• Hierarchical Models — the primary use case where MCMC is indispensable