Resonances in reflective Hamiltonian Monte Carlo

Abstract

In high dimensions, reflective Hamiltonian Monte Carlo with inexact reflections exhibits slow mixing when the particle ensemble is initialised from a Dirac delta distribution and the uniform distribution is targeted. By quantifying the instantaneous non-uniformity of the distribution with the Sinkhorn divergence, we elucidate the principal mechanisms underlying the mixing problems. In spheres and cubes, we show that the collective motion transitions between fluid-like and discretisation-dominated behaviour, with the critical step size scaling as a power law in the dimension. In both regimes, the particles can spontaneously unmix, leading to resonances in the particle density and the aforementioned problems. Additionally, low-dimensional toy models of the dynamics are constructed which reproduce the dominant features of the high-dimensional problem. Finally, the dynamics is contrasted with the exact Hamiltonian particle flow and tuning practices are discussed.

References in corpus (10)

• MCMC using Hamiltonian dynamics
• MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics
• Ab initio Random Structure Searching
• PolyChord: next-generation nested sampling
• Nested sampling for physical scientists
• Nested Sampling Methods
• Bayesian model selection without evidences: application to the dark energy equation-of-state
• Analytical marginalisation over photometric redshift uncertainties in cosmic shear analyses
• Nested sampling with plateaus
• Kernel-, mean- and noise-marginalised Gaussian processes for exoplanet transits and $H_0$ inference