Gibbs Sampler for Matrix Generalized Inverse Gaussian Distributions

Abstract

Sampling from matrix generalized inverse Gaussian (MGIG) distributions is required in Markov chain Monte Carlo (MCMC) algorithms for a variety of statistical models. However, an efficient sampling scheme for the MGIG distributions has not been fully developed. We here propose a novel blocked Gibbs sampler for the MGIG distributions based on the Cholesky decomposition. We show that the full conditionals of the entries of the diagonal and unit lower-triangular matrices are univariate generalized inverse Gaussian and multivariate normal distributions, respectively. Several variants of the Metropolis-Hastings algorithm can also be considered for this problem, but we mathematically prove that the average acceptance rates become extremely low in particular scenarios. We demonstrate the computational efficiency of the proposed Gibbs sampler through simulation studies and data analysis. Supplementary materials for this article are available online.

KEYWORDS:

• Matrix generalized inverse Gaussian distributions
• Matrix skew-t distributions
• Markov chain Monte Carlo
• Partial Gaussian graphical models

Supplementary Materials

Supplementary Materials for “Gibbs Sampler for Matrix Generalized Inverse Gaussian Distributions”: This Supplementary Materials provide theoretical details of the main document and additional simulation results. (pdf)

Acknowledgments

We would like to thank the editor, the associate editor, and the two reviewers for many valuable comments and helpful suggestions that led to an improved version of this article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Research of the authors was supported in part by JSPS KAKENHI Grant Number 22K20132, 19K11852, 17K17659, and 21H00699 from Japan Society for the Promotion of Science.