🤖 AI Summary
This work addresses the challenge of efficient sampling from finite mixtures of Boltzmann–Gibbs distributions by proposing a switching Hamiltonian Monte Carlo method that accurately simulates state-switching dynamics through the coupling of a symmetric numerical integrator with a Poisson jump process. Theoretically, it establishes for the first time that the integrator incurs a second-order bias and develops an error estimation framework based on the discrete Poisson equation. Furthermore, geometric ergodicity of the associated Markov chain is rigorously proven. Experimental results confirm that the method’s convergence rate aligns with theoretical predictions, significantly enhancing both the efficiency and accuracy of sampling from mixture distributions.
📝 Abstract
We introduce a switching Hamiltonian Monte Carlo method for sampling from finite mixture Boltzmann-Gibbs distributions. We propose symmetric numerical integrators to approximate switching Hamiltonian dynamics interlaced with Poisson jumps, where the regime-switching chain is simulated using the uniformization technique or the stochastic simulation algorithm. We prove geometric ergodicity of the resulting Markov chain. We develop an approach based on the discrete Poisson equation associated with numerical schemes to estimate the error in computing ergodic averages. Using this approach we prove that the proposed numerical integrators have second-order bias. This approach is simple and can be generalized to other settings, for example, kinetic Langevin equations. Finally, we verify the convergence result via numerical experiment.