Multiple Jump MCMC: A Scalable Algorithm for Bayesian Inference on Binary Model Spaces

TL;DR

This paper introduces Multiple Jump MCMC, a rejection-free, fast algorithm for Bayesian inference on binary model spaces, significantly outperforming existing methods especially in high-dimensional Gaussian graphical models.

Contribution

The paper presents a novel, simple, rejection-free MCMC algorithm that achieves much faster exploration of binary model spaces compared to traditional methods.

Findings

Multiple Jump MCMC is 100-200 times faster than state-of-the-art methods.

The algorithm accurately targets the posterior distribution.

It applies broadly to Gaussian graphical models, Ising models, and variable selection.

Abstract

This article considers Bayesian model inference on binary model spaces. Binary model spaces are used by a large class of models, including graphical models, variable selection, mixture distributions, and decision trees. Traditional strategies in this field, such as reversible jump or birth-death MCMC algorithms, are still popular, despite suffering from a slow exploration of the model space. In this article, we propose an alternative: the Multiple Jump MCMC algorithm. The algorithm is simple, rejection-free, and remarkably fast. When applied to undirected Gaussian graphical models, it is 100 to 200 times faster than the state-of-the-art, solving models with $500,000$ parameters in less than a minute. We provide theorems showing how accurately our algorithm targets the posterior, and we show how to apply our framework to Gaussian graphical models, Ising models, and variable selection, but note that it applies to most Bayesian posterior inference on binary model spaces.