Exploiting Exact Conditionals Improves Conditioning: Provably Fast Mixing Time Bounds By Sampling from the Marginal

Abstract

The problem of sampling from a probability distribution arises in many applications such as posterior sampling in hierarchical Bayesian inverse problems and Gaussian processes for machine learning. Markov chain Monte Carlo (MCMC) algorithms are often used for sampling from a target probability distribution, but implementations can be computationally expensive, especially for large-scale problems. In certain applications, the target distribution naturally factorizes into a lower dimensional marginal distribution and a conditional distribution that allows exact sampling. We describe an MCMC algorithm called MarCo that exploits such a structure and generates a Markov chain via Metropolis-Hastings sampling from the marginal distribution, followed by sampling from the exact conditional distribution. By design, MarCo constructs a Markov chain on the joint space that inherits the convergence behavior of the marginal MCMC algorithm. This provides multiple theoretical and computational advantages. We prove that MarCo can achieve improved mixing time upper bounds compared to direct sampling from the joint distribution. Moreover, compared to one-block methods that also exploit marginal-conditional structure, we use the framework of Peskun-Tierney ordering to show that MarCo has a larger right spectral gap and smaller asymptotic variance, thus leading to superior convergence properties. Numerical results illustrate the performance benefits of MarCo and are provided for various problems, including a semi-blind image deblurring example.