Graph-Enabled Fast MCMC Sampling with an Unknown High-Dimensional Prior Distribution

TL;DR

This paper introduces a graph-enabled MCMC method that efficiently samples from posteriors with unknown, high-dimensional priors by leveraging a geometric graph constructed from prior samples, demonstrating both theoretical soundness and computational efficiency.

Contribution

The novel graph-enabled MCMC algorithm effectively handles unknown high-dimensional priors, improving posterior sampling efficiency in Bayesian inference.

Findings

Provides reliable approximation to the posterior distribution.

Demonstrates high computational efficiency.

Validates effectiveness through theoretical and numerical studies.

Abstract

Posterior sampling is a task of central importance in Bayesian inference. For many applications in Bayesian meta-analysis and Bayesian transfer learning, the prior distribution is unknown and needs to be estimated from samples. In practice, the prior distribution can be high-dimensional, adding to the difficulty of efficient posterior inference. In this paper, we propose a novel Markov chain Monte Carlo algorithm, which we term graph-enabled MCMC, for posterior sampling with unknown and potentially high-dimensional prior distributions. The algorithm is based on constructing a geometric graph from prior samples and subsequently uses the graph structure to guide the transition of the Markov chain. Through extensive theoretical and numerical studies, we demonstrate that our graph-enabled MCMC algorithm provides reliable approximation to the posterior distribution and is highly computationally efficient.