The Soft Multivariate Truncated Normal Distribution with Applications to Bayesian Constrained Estimation

TL;DR

The paper introduces the soft tMVN distribution, a smooth approximation to the truncated multivariate normal, along with an efficient sampling method and theoretical validation, useful for Bayesian constrained estimation.

Contribution

It proposes the soft tMVN distribution, develops a Gibbs sampler for high-dimensional sampling, and provides theoretical and empirical validation of the approximation.

Findings

The soft tMVN effectively approximates the tMVN distribution.

The Gibbs sampler efficiently samples in high dimensions.

The method is applicable to Bayesian shape-constrained problems.

Abstract

We propose a new distribution, called the soft tMVN distribution, which provides a smooth approximation to the truncated multivariate normal (tMVN) distribution with linear constraints. An efficient blocked Gibbs sampler is developed to sample from the soft tMVN distribution in high dimensions. We provide theoretical support to the approximation capability of the soft tMVN and provide further empirical evidence thereof. The soft tMVN distribution can be used to approximate simulations from a multivariate truncated normal distribution with linear constraints, or itself as a prior in shape-constrained problems.