A class of sampling algorithms that constructs a Markov chain whose stationary distribution equals a target probability distribution, producing dependent samples used to approximate expectations and probability integrals for complex or high-dimensional distributions.

Principle

Build a transition kernel that satisfies detailed balance or ergodicity with respect to the target distribution so that samples drawn along the chain asymptotically follow that distribution; practical implementations enforce mixing, burn-in, and thinning policies to mitigate dependence and initialization bias.

Demonstration

Use the Metropolis-Hastings algorithm to sample a Bayesian posterior for a hierarchical model: propose a parameter update, accept with probability ratio of target densities multiplied by proposal correction, and repeat to obtain a sample chain whose empirical averages approximate posterior means.

Misapplication

Treating correlated MCMC draws as independent in variance estimation or using insufficient burn-in and treating early samples as representative, which underestimates uncertainty and yields biased parameter summaries.

Consequence

When correctly implemented and diagnosed, MCMC provides asymptotically correct approximations of expectations, credible intervals, and marginal densities for models that are analytically intractable, enabling principled Bayesian inference in high dimensions.

Reversal

Independent sampling methods (exact i.i.d. sampling or importance sampling with independent proposals) produce uncorrelated draws and can avoid chain dependence at the cost of requiring tractable proposals or excessive variance in high dimensions.

Boundary

Applies to target distributions where a suitable Markov kernel can be constructed and mixed in reasonable time; it does not guarantee quick convergence for multimodal, high-curvature, or extremely high-dimensional targets without advanced samplers or tuning.

Semantic Tension

Tension exists between MCMC and variational inference or importance sampling: MCMC targets asymptotic exactness via dependence, while variational methods trade exactness for speed via independent approximate distributions; both aim to approximate the same targets but with different error modes.

Synthesis

MCMC is a dependent-sample framework that transforms local transition rules into global approximations of difficult probability distributions; its power lies in constructing ergodic kernels whose long-run empirical measures converge to the target, but practical success depends on diagnostics, tuning, and computational resources.