An Auxiliary Variable Trick for MCMC

[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 from arbitrary continuous probability distributions. We could simply factor the distribution $\pi({\bf x})$ as


\pi({\bf x}) = \mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma) \cdot \frac{\pi({\bf x})}{\mathcal N({\bf x};\boldsymbol\mu,\boldsymbol\Sigma)},

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