Design of Hamiltonian Monte Carlo for perfect simulation of general continuous distributions

Abstract

Hamiltonian Monte Carlo (HMC) is an efficient method of simulating smooth distributions and has motivated the widely used No-U-turn Sampler (NUTS) and software Stan. We build on NUTS and the technique of "unbiased sampling" to design HMC algorithms that produce perfect simulation of general continuous distributions that are amenable to HMC. Our methods enable separation of Markov chain Monte Carlo convergence error from experimental error, and thereby provide much more powerful MCMC convergence diagnostics than current state-of-the-art summary statistics which confound these two errors. Objective comparison of different MCMC algorithms is provided by the number of derivative evaluations per perfect sample point. We demonstrate the methodology with applications to normal, $t$ and normal mixture distributions up to 100 dimensions, and a 12-dimensional Bayesian Lasso regression. HMC runs effectively with a goal of 20 to 30 points per trajectory. Numbers of derivative evaluations per perfect sample point range from 390 for a univariate normal distribution to 12,000 for a 100-dimensional mixture of two normal distributions with modes separated by six standard deviations, and 22,000 for a 100-dimensional $t$-distribution with four degrees of freedom.