## Pseudo-marginal MCMC

Robert Nishihara

[latexpage]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 $\pi({\bf x})$, with ${\bf x} \in \mathcal X$, 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 ${\bf x}$ to a proposed state ${\bf x}’$ with probability $\min(1, \pi({\bf x}’)/\pi({\bf x}))$ . But what if we cannot evaluate $\pi({\bf x})$ exactly? Such a situation might arise if we are given a joint density function $\pi({\bf x}, {\bf z})$, with ${\bf z} \in \mathcal Z$, and we must …

## An Auxiliary Variable Trick for MCMC

Robert Nishihara 1 Comment

[latexpage]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 $\pi({\bf x})$ (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 $$\pi({\bf x}) \propto \mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma) L({\bf x}),$$ where $\mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma)$ is a multivariate Gaussian prior with mean $\boldsymbol\mu$ and covariance matrix $\boldsymbol\Sigma$, and $L({\bf x})$ is a likelihood function. Suppose we want to generalize this algorithm to sample …