Markov Chain Monte Carlo

Many posterior distributions cannot be sampled from directly using a simple independent-draw method. MCMC builds a Markov chain whose long-run stationary distribution is the target. Each new state depends on the current state, and under suitable conditions, averages across draws can approximate posterior expectations. Because draws are dependent, the number of saved rows is not the same as the amount of independent information. In Metropolis-Hastings, a proposal distribution suggests a candidate state. The algorithm accepts that candidate with a probability based on the ratio of target densities and, when the proposal is asymmetric, a correction for forward and reverse proposal probabilities. If it rejects, the chain repeats the previous state. A symmetric random-walk proposal simplifies the ratio, but a poorly scaled proposal may move too rarely or take inefficient steps. Gibbs sampling is a special conditional-update approach: sample each variable from its full conditional given the current values of the others. It works when those conditional distributions are available and the chain has suitable properties. Imagine a one-dimensional target where proposals are often rejected because step sizes are too large. The chain repeats states, increasing autocorrelation and lowering effective sample size. Very small steps may accept frequently but explore slowly. There is a tradeoff between acceptance and movement, and diagnostics should examine whether different chains explore the same regions. Discarding initial warmup can reduce dependence on initialization, but there is no fixed number of draws that guarantees convergence. Run multiple chains from dispersed starting points, inspect trace plots, and examine R-hat and effective sample size for quantities that matter. These diagnostics can fail to reveal all pathologies, particularly with separated modes or poorly explored tails. Also inspect sampler-specific warnings and Monte Carlo error. MCMC estimates numerical uncertainty from finite simulation; they do not remove uncertainty in the statistical model or data. Report what was sampled, how it was diagnosed and which posterior summaries are reliable enough for the decision.