MetropolisHastingstype

Metropolis-Hastings type refers to a family of Markov chain Monte Carlo methods that generate samples from a target distribution by constructing a Markov chain with that distribution as its stationary distribution. The framework generalizes the original Metropolis algorithm by allowing a broad class of proposal distributions, including non-symmetric ones, which broadens its applicability in Bayesian inference, statistics, and statistical physics. The generalization was formalized by W. K. Hastings in 1970, showing how to adjust for proposal asymmetry so the chain still targets the desired distribution.

Algorithmically, the process starts at a current state x. A proposed state x' is drawn from a proposal distribution q(x'|x). The acceptance probability is alpha = min(1, [π(x') q(x|x')] / [π(x) q(x'|x)]), where π is the unnormalized target density (often a posterior density in Bayesian work). With probability alpha, the chain moves to x'; otherwise it remains at x. Repeating this step yields a Markov chain whose stationary distribution is π under standard regularity conditions (irreducibility and aperiodicity).

Choices of the proposal distribution q influence convergence and mixing. Common variants include symmetric random-walk proposals, independent proposals, and gradient-informed proposals such as the Metropolis-adjusted Langevin algorithm (MALA). In practice, only π up to a normalizing constant is needed, which makes MH-type methods well suited to posterior sampling in complex models.

Historically, the Metropolis algorithm (1953) used symmetric proposals, while Hastings’ extension (1970) allowed non-symmetric proposals, giving rise to the common designation of Metropolis-Hastings type methods. Today, MH-type methods are foundational in Bayesian computation and wide-ranging Monte Carlo applications.