This post gives a brief introduction to the pseudo-marginal approach to MCMC. A very nice explanation, with examples, is available here. Frequently, we are given a density function , with , andÂ we use Markov chain Monte Carlo (MCMC) to generate samples from the corresponding probability distribution. For simplicity, suppose we are performing Metropolis-Hastings with a spherical proposal distribution. Then, we move from the current state to a proposed state with probability . But what if we cannot evaluate exactly? Such a situation might arise if we are given a joint density function , with , and we must marginalize out in order to compute . In this situation, we may only be able to approximate