Continuously-Tempered PDMP Samplers

TL;DR

This paper introduces a tempering approach to enhance the mixing of PDMP samplers, especially for multi-modal and heavy-tailed distributions, by interpolating between tractable and target distributions using an inverse temperature parameter.

Contribution

The authors propose a novel tempering method integrated with PDMPs, particularly the Zig-Zag sampler, to improve sampling efficiency for complex distributions.

Findings

Outperforms existing PDMP samplers on multimodal posteriors

Easy to implement with demonstrated empirical improvements

Effectively explores distributions at multiple temperature levels

Abstract

New sampling algorithms based on simulating continuous-time stochastic processes called piece-wise deterministic Markov processes (PDMPs) have shown considerable promise. However, these methods can struggle to sample from multi-modal or heavy-tailed distributions. We show how tempering ideas can improve the mixing of PDMPs in such cases. We introduce an extended distribution defined over the state of the posterior distribution and an inverse temperature, which interpolates between a tractable distribution when the inverse temperature is 0 and the posterior when the inverse temperature is 1. The marginal distribution of the inverse temperature is a mixture of a continuous distribution on [0,1) and a point mass at 1: which means that we obtain samples when the inverse temperature is 1, and these are draws from the posterior, but sampling algorithms will also explore distributions at lower temperatures which will improve mixing. We show how PDMPs, and particularly the Zig-Zag sampler, can be implemented to sample from such an extended distribution. The resulting algorithm is easy to implement and we show empirically that it can outperform existing PDMP-based samplers on challenging multimodal posteriors.