Sampling From Multiscale Densities With Delayed Rejection Generalized Hamiltonian Monte Carlo

Abstract

With the increasing prevalence of probabilistic programming languages, Hamiltonian Monte Carlo (HMC) has become the mainstay of applied Bayesian inference. However HMC still struggles to sample from densities with multiscale geometry: a large step size is needed to efficiently explore low curvature regions while a small step size is needed to accurately explore high curvature regions. We introduce the delayed rejection generalized HMC (DR-G-HMC) sampler that overcomes this challenge by employing dynamic step size selection, inspired by differential equation solvers. In a single sampling iteration, DR-G-HMC sequentially makes proposals with geometrically decreasing step sizes if necessary. This simulates Hamiltonian dynamics with increasing fidelity that, in high curvature regions, generates proposals with a higher chance of acceptance. DR-G-HMC also makes generalized HMC competitive by decreasing the number of rejections which otherwise cause inefficient backtracking and prevents directed movement. We present experiments to demonstrate that DR-G-HMC (1) correctly samples from multiscale densities, (2) makes generalized HMC methods competitive with the state of the art No-U-Turn sampler, and (3) is robust to tuning parameters.