## Pseudo-marginal MCMC

Robert Nishihara

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 , and 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

## An Auxiliary Variable Trick for MCMC

Robert Nishihara1 Comment

I recently uploaded the paper “Parallel MCMC with Generalized Elliptical Slice Sampling” to the arXiv. I’d like to highlight one trick that we used, but first I’ll give some background. Markov chain Monte Carlo (MCMC) is a class of algorithms for generating samples from a specified probability distribution (in the continuous setting, the distribution is generally specified by its density function). Elliptical slice sampling is an MCMC algorithm that can be used to sample distributions of the form (1)   where is a multivariate Gaussian prior with mean and covariance matrix , and is a likelihood function. Suppose we want to generalize this algorithm to sample from arbitrary continuous probability distributions. We could simply factor the distribution as (2)