Metropolis–Hastings explained: the MCMC engine behind Bayesian inference in ML
burny_tech · x · 2026-09-20
A well-crafted explainer thread on the Metropolis–Hastings (MH) algorithm, the MCMC method for sampling from complex probability distributions when direct sampling is hard.
Key mechanics: propose a new state x′ from a proposal distribution q(x′|x) and accept it with probability α = min{1, [π(x′)q(x|x′)]/[π(x)q(x′|x)]}; rejections keep the chain in place. Under suitable conditions the chain's stationary distribution is π, so sample averages approximate expectations under the target.
In ML, when the Bayesian posterior p(θ|D) is analytically intractable, MCMC yields approximate posterior samples for estimating parameters, credible intervals and predictive distributions — foundational to Bayesian inference.
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