Sampling constrained continuous probability distributions: A review

Abstract

The problem of sampling constrained continuous distributions has frequently appeared in many machine/statistical learning models. Many Markov Chain Monte Carlo (MCMC) sampling methods have been adapted to handle different types of constraints on random variables. Among these methods, Hamilton Monte Carlo (HMC) and the related approaches have shown significant advantages in terms of computational efficiency compared with 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 with constrained design space.This article is categorized under: Applications of Computational Statistics > Computational Mathematics Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical and Graphical Methods of Data Analysis > Markov Chain Monte Carlo (MCMC) Statistical and Graphical Methods of Data Analysis > Sampling

Convergence Rates for the Constrained Sampling via Langevin Monte Carlo

“…Firstly, based on MCMC algorithms, a direct solution involves discarding samples that violate the constraints, thereby exclusively retaining samples that satisfy the constraints; see, for example, [ 1 , 15 , 16 ]. However, these rejection-type approaches may encounter an excessive number of rejections or an extremely large acceptance rate within some local subspace that satisfies the constraints, which leads to poor mixing and computational inefficiency, especially for complicated constraints and the high dimensional distributions [ 17 , 18 ]. Secondly, some literature draws inspiration from penalty functions in optimization problems and considers the construction of barriers along the boundaries of the constrained domain, effectively constraining the sampling process within the constrained area.…”