Gibbs Sampler for Matrix Generalized Inverse Gaussian Distributions

TL;DR

This paper introduces a new blocked Gibbs sampler for matrix generalized inverse Gaussian distributions, improving sampling efficiency in MCMC algorithms for statistical models.

Contribution

A novel Gibbs sampling method based on Choleski decomposition for MGIG distributions, with proven efficiency and better acceptance rates than Metropolis-Hastings variants.

Findings

The Gibbs sampler efficiently samples from MGIG distributions.

Acceptance rates for Metropolis-Hastings are very low in certain cases.

Simulation studies show the proposed method's computational advantages.

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 Choleski decomposition. We show that the full conditionals of the diagonal and unit lower-triangular entries 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.