A Piecewise Deterministic Markov Process via $(r,\theta)$ swaps in hyperspherical coordinates

TL;DR

This paper introduces a novel nonlinear transition function for piecewise deterministic Markov processes using hyperspherical coordinates, enhancing their ability to target arbitrary distributions for efficient Bayesian inference.

Contribution

It derives a new class of PDMPs with nonlinear transitions, expanding the theoretical framework and practical applicability of these processes in statistical computation.

Findings

Effective on Gaussian targets

Applied successfully to Bayesian logistic regression

Provides insights into PDMP theory and applications

Abstract

Recently, a class of stochastic processes known as piecewise deterministic Markov processes has been used to define continuous-time Markov chain Monte Carlo algorithms with a number of attractive properties, including compatibility with stochastic gradients like those typically found in optimization and variational inference, and high efficiency on certain big data problems. Not many processes in this class that are capable of targeting arbitrary invariant distributions are currently known, and within one subclass all previously known processes utilize linear transition functions. In this work, we derive a process whose transition function is nonlinear through solving its Fokker-Planck equation in hyperspherical coordinates. We explore its behavior on Gaussian targets, as well as a Bayesian logistic regression model with synthetic data. We discuss implications to both the theory of piecewise deterministic Markov processes, and to Bayesian statisticians as well as physicists seeking to use them for simulation-based computation.