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A generalized discontinuous Hamilton Monte Carlo for transdimensional sampling

Lei Li, Xiangxian Luo, Yinchen Luo

TL;DR

This work extends discontinuous Hamiltonian Monte Carlo (DHMC) to trans-dimensional sampling, addressing distributions with variable dimension such as the grand canonical ensemble. By embedding the dimension index into a continuous variable and introducing a measure-transform-based expansion for added coordinates along with an energy-correction term, the authors preserve detailed balance and enable efficient sampling across dimensions; the method naturally interprets the dimension-change cost as a free-energy contribution. The framework includes a discretized, RJMCMC-compatible implementation and proves detailed balance for the ideal algorithm, with practical discretization realized via operator splitting and random-time evolution. When combined with the Random Batch Method, the approach achieves scalable $O(N)$ per-iteration cost for interacting particle systems, demonstrated through 1D free-gas, cosine-potential, and Lennard-Jones examples, with reduced autocorrelation relative to Metropolis-Hastings. Overall, the generalized DHMC provides a principled, efficient tool for simulating trans-dimensional targets like the grand canonical ensemble in both non-interacting and interacting regimes, with broad applicability to Bayesian model selection and particle systems.

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.

A generalized discontinuous Hamilton Monte Carlo for transdimensional sampling

TL;DR

This work extends discontinuous Hamiltonian Monte Carlo (DHMC) to trans-dimensional sampling, addressing distributions with variable dimension such as the grand canonical ensemble. By embedding the dimension index into a continuous variable and introducing a measure-transform-based expansion for added coordinates along with an energy-correction term, the authors preserve detailed balance and enable efficient sampling across dimensions; the method naturally interprets the dimension-change cost as a free-energy contribution. The framework includes a discretized, RJMCMC-compatible implementation and proves detailed balance for the ideal algorithm, with practical discretization realized via operator splitting and random-time evolution. When combined with the Random Batch Method, the approach achieves scalable per-iteration cost for interacting particle systems, demonstrated through 1D free-gas, cosine-potential, and Lennard-Jones examples, with reduced autocorrelation relative to Metropolis-Hastings. Overall, the generalized DHMC provides a principled, efficient tool for simulating trans-dimensional targets like the grand canonical ensemble in both non-interacting and interacting regimes, with broad applicability to Bayesian model selection and particle systems.

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.
Paper Structure (17 sections, 2 theorems, 84 equations, 7 figures, 4 algorithms)

This paper contains 17 sections, 2 theorems, 84 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. A Brief Review of the DHMC
  3. Generalized Discrete Hamilton Monte Carlo for Trans-dimensional Sampling
  4. The effective Hamiltonian
  5. The Generalized Discontinuous Hamiltonian Monte Carlo
  6. Theoretical properties of DHMC
  7. Discretization of DHMC and discussion
  8. Discretized DHMC
  9. DHMC as a special realization of RJMCMC
  10. DHMC for grand canonical ensemble
  11. DHMC for the Grand Canonical Ensemble
  12. Combining DHMC with Random Batch Method
  13. Numerical Examples
  14. A one dimensional free gas model
  15. A system with an artificial cosine potential
  16. ...and 2 more sections

Key Result

Lemma 3.1

Assume that $\pi_{N,N+1}(\mathbf{q}_{\mathrm{new}}, \mathbf{p}_{\mathrm{new}}; \mathbf{q}, \mathbf{p})$ is even in $\mathbf{p}$ and $\mathbf{p}_{\mathrm{new}}$. For evolution time $T<1$, the transition kernel of DHMC dynamics satisfies the detailed balance condition.

Figures (7)

  • Figure 1: Left panel: Empirical distribution of particle numbers for DHMC samplers with a sample size of $9\times10^5$; Right panel: Error versus different sample sizes in log scale, with a reference line having a slope of -1/2
  • Figure 2: Left panel: Empirical distributions of particle numbers obtained using DHMC samplers with varying sample sizes. The reference distribution is given by the empirical distribution of the MH sampler with a sample size of $10^8$ at equilibrium. Right panel: Error versus different sample sizes, with a reference line having a slope of -1/2
  • Figure 3: The line is the equation of state proposed by Johnsonjohnson1993lennard and the marks are simulation results given by Metropolis-Hastings algorithm and our RB-DHMC.
  • Figure 4: The CPU time of RB-DHMC grows linearly with respect to the number of particles.
  • Figure 5: Upper: The pressure evaluated by RB-DHMC. Lower: The pressure evaluated by MH.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Remark 5.1