Spherical Hamiltonian Monte Carlo for Constrained Target Distributions

TL;DR

This paper introduces a novel MCMC method that maps constrained parameter spaces onto spheres, enabling efficient sampling by leveraging geodesic flows on the sphere to handle constraints naturally.

Contribution

The paper presents a new spherical Hamiltonian Monte Carlo approach that maps constrained domains to spheres and exploits geodesic flows for improved sampling efficiency.

Findings

Effective handling of various constraints in target distributions.

Improved sampling efficiency demonstrated on multiple models.

Natural framework for constrained Bayesian inference.

Abstract

We propose a new Markov Chain Monte Carlo (MCMC) method for constrained target distributions. Our method first maps the $D$-dimensional constrained domain of parameters to the unit ball ${\bf B}_0^D(1)$. Then, it augments the resulting parameter space to the $D$-dimensional sphere, ${\bf S}^D$. The boundary of ${\bf B}_0^D(1)$ corresponds to the equator of ${\bf S}^D$. This change of domains enables us to implicitly handle the original constraints because while the sampler moves freely on the sphere, it proposes states that are within the constraints imposed on the original parameter space. To improve the computational efficiency of our algorithm, we split the Lagrangian dynamics into several parts such that a part of the dynamics can be handled analytically by finding the geodesic flow on the sphere. We apply our method to several examples including truncated Gaussian, Bayesian Lasso, Bayesian bridge regression, and a copula model for identifying synchrony among multiple neurons. Our results show that the proposed method can provide a natural and efficient framework for handling several types of constraints on target distributions.