Introduction

Introduction

Bayesian inference needs samples from a posterior $p(\boldsymbol{\theta} \mid \mathcal{D}) \propto p(\mathcal{D} \mid \boldsymbol{\theta})p(\boldsymbol{\theta})$ that we can only evaluate up to a constant, often in many dimensions. Importance sampling breaks down there. Markov chain Monte Carlo (MCMC) instead builds a random walk that, after a while, visits each region in proportion to its probability. Its samples can then be averaged like ordinary Monte Carlo samples.