Sampling Constrained Continuous Probability Distributions: A Review

TL;DR

This review discusses advanced Hamilton Monte Carlo methods for sampling from constrained continuous distributions, highlighting their efficiency and applications in various statistical problems.

Contribution

It provides a comprehensive review of HMC-based constrained sampling methods and introduces three specific techniques with practical applications.

Findings

HMC methods are efficient for constrained sampling.

Three constrained HMC techniques are explained: reflection, reformulation, spherical HMC.

Applications include truncated normal, Bayesian regression, and density estimation.

Abstract

The problem of sampling constrained continuous distributions has frequently appeared in many machine/statistical learning models. Many Monte Carlo Markov Chain (MCMC) sampling methods have been adapted to handle different types of constraints on the random variables. Among these methods, Hamilton Monte Carlo (HMC) and the related approaches have shown significant advantages in terms of computational efficiency compared to other counterparts. In this article, we first review HMC and some extended sampling methods, and then we concretely explain three constrained HMC-based sampling methods, reflection, reformulation, and spherical HMC. For illustration, we apply these methods to solve three well-known constrained sampling problems, truncated multivariate normal distributions, Bayesian regularized regression, and nonparametric density estimation. In this review, we also connect constrained sampling with another similar problem in the statistical design of experiments of constrained design space.