Markov Chain Monte Carlo (MCMC) is a class of algorithms used for sampling from probability distributions when direct sampling is challenging. It combines principles from Markov chains and Monte Carlo methods to allow for the estimation of complex distributions, particularly in high-dimensional spaces. ### Key Concepts: 1. **Markov Chain**: A Markov chain is a sequence of random variables where the distribution of the next variable depends only on the current variable and not on the previous states (the Markov property).
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Markov chain Monte Carlo estimates expectations under a target probability distribution by simulating a Markov chain that preserves . For an integrable observable , the assumptions of a Markov-chain law of large numbers justifyalmost surely. A Gibbs sampler preserves the target through full conditional distributions; a Metropolis–Hastings algorithm proposes a move and accepts or rejects it using a ratio of target and proposal densities. The unknown normalization constant cancels from that ratio, which makes unnormalized Bayesian posterior densities usable.
Successive states are generally dependent, so Monte Carlo precision depends on the observable's integrated autocorrelation time, rather than the raw iteration count alone. Burn-in can reduce initial-state effects but is not a proof of convergence. A correct stationary distribution also does not ensure rapid exploration: isolated modes or strong dependence can produce a long mixing time. Markov chain Monte Carlo convergence diagnostics investigate these limitations.