A note on simulation methods for the Dirichlet-Laplace prior

TL;DR

This paper identifies an issue in the original MCMC sampling algorithm for the Dirichlet-Laplace prior and proposes corrected and alternative algorithms to ensure proper posterior sampling, without affecting the original theoretical results.

Contribution

It corrects the sampling procedure for the Dirichlet-Laplace prior and introduces a new, equivalent algorithm to improve posterior inference accuracy.

Findings

Corrected the original MCMC sampling algorithm.

Proposed a new, equivalent sampling algorithm.

Ensured proper joint posterior sampling for the DL prior.

Abstract

Bhattacharya et al. (2015, Journal of the American Statistical Association 110(512): 1479-1490) introduce a novel prior, the Dirichlet-Laplace (DL) prior, and propose a Markov chain Monte Carlo (MCMC) method to simulate posterior draws under this prior in a conditionally Gaussian setting. The original algorithm samples from conditional distributions in the wrong order, i.e., it does not correctly sample from the joint posterior distribution of all latent variables. This note details the issue and provides two simple solutions: A correction to the original algorithm and a new algorithm based on an alternative, yet equivalent, formulation of the prior. This corrigendum does not affect the theoretical results in Bhattacharya et al. (2015).