Orbital MCMC

TL;DR

This paper introduces a new acceptance test for stochastic Markov kernels in MCMC, allowing for more flexible proposal transformations, and presents two practical algorithms with demonstrated empirical benefits.

Contribution

It derives a generalized acceptance test for stochastic kernels and proposes two novel MCMC algorithms based on periodic orbits and state space contractions.

Findings

Empirical results show improved sampling efficiency

Proposed algorithms outperform traditional MCMC methods

Flexible proposal transformations enhance convergence

Abstract

Markov Chain Monte Carlo (MCMC) algorithms ubiquitously employ complex deterministic transformations to generate proposal points that are then filtered by the Metropolis-Hastings-Green (MHG) test. However, the condition of the target measure invariance puts restrictions on the design of these transformations. In this paper, we first derive the acceptance test for the stochastic Markov kernel considering arbitrary deterministic maps as proposal generators. When applied to the transformations with orbits of period two (involutions), the test reduces to the MHG test. Based on the derived test we propose two practical algorithms: one operates by constructing periodic orbits from any diffeomorphism, another on contractions of the state space (such as optimization trajectories). Finally, we perform an empirical study demonstrating the practical advantages of both kernels.