An Intuitive Guide to MCMC (Part I): The Metropolis-Hastings Algorithm

A detailed walkthrough of Markov Chain Monte Carlo (MCMC) methods, starting with the Metropolis-Hastings algorithm. Covers the core problem of sampling from unnormalized distributions, explains Markov chain theory including stationary distributions and ergodicity, derives the Metropolis-Hastings acceptance criterion and the Hastings correction, and demonstrates Python implementations on a Gaussian and a 2D 'volcano' distribution. The key insight is that the normalization constant cancels in the acceptance ratio, enabling sampling without solving intractable integrals.