A generalized discontinuous Hamilton Monte Carlo for transdimensional sampling

TL;DR

This paper introduces a generalized discontinuous Hamilton Monte Carlo (DHMC) method for efficient sampling from transdimensional distributions, especially the grand canonical ensemble, by incorporating measure transforms and energy corrections.

Contribution

It extends DHMC to variable dimensions with a measure transform approach, enabling natural sampling of the grand canonical ensemble with improved efficiency.

Findings

DHMC with measure transform satisfies detailed balance.

The combined DHMC and random batch method reduces sample correlation.

Experimental results outperform traditional Metropolis-Hastings in efficiency.

Abstract

In this paper, we propose a discontinuous Hamilton Monte Carlo (DHMC) to sample from dimensional varying distributions, and particularly the grand canonical ensemble. The DHMC was proposed in [Biometrika, 107(2)] for discontinuous potential where the variable has a fixed dimension. When the dimension changes, there is no clear explanation of the volume-preserving property, and the conservation of energy is also not necessary. We use a random sampling for the extra dimensions, which corresponds to a measure transform. We show that when the energy is corrected suitably for the trans-dimensional Hamiltonian dynamics, the detailed balance condition is then satisfied. For the grand canonical ensemble, such a procedure can be explained very naturally to be the extra free energy change brought by the newly added particles, which justifies the rationality of our approach. To sample the grand canonical ensemble for interacting particle systems, the DHMC is then combined with the random batch method to yield an efficient sampling method. In experiments, we show that the proposed DHMC combined with the random batch method generates samples with much less correlation when compared with the traditional Metropolis-Hastings method.