MCMC

MCMC, or Markov chain Monte Carlo, is a class of algorithms for sampling from complex probability distributions by constructing a Markov chain whose stationary distribution is the target distribution. The Monte Carlo aspect uses random sampling to estimate expectations, while the Markov chain aspect provides a mechanism to explore high‑dimensional state spaces when direct sampling is impractical.

In Bayesian statistics, MCMC is widely used to approximate posterior distributions that arise when closed-form solutions are unavailable. The typical approach is to design a transition kernel that preserves the target distribution, often via detailed balance. After a burn-in period, the chain is assumed to have reached its stationary distribution, and the collected samples are used to estimate quantities such as means, variances, credible intervals, and predictive distributions.

Common algorithms include Metropolis-Hastings, which proposes new states and accepts them with a probability ensuring reversibility, and Gibbs sampling, which draws successive values from full conditional distributions. Variants such as Hamiltonian Monte Carlo (HMC) use gradient information to improve efficiency, with No-U-Turn Sampler (NUTS) automating tuning. Other methods include slice sampling and adaptive MCMC, which tunes proposals on the fly. Convergence and efficiency are assessed with diagnostics such as trace plots, autocorrelation, effective sample size, and the Gelman‑Rubin statistic (R-hat).

Key considerations in practice include choosing proposal distributions, diagnosing convergence and mixing, determining burn-in and thinning, and balancing computational cost with accuracy. MCMC supports posterior inference and high‑dimensional integration across fields such as statistics, physics, machine learning, and computational biology.