On Data Augmentation for Models Involving Reciprocal Gamma Functions

TL;DR

This paper presents a novel data augmentation method for models involving reciprocal gamma functions, enabling efficient posterior inference through approximations and advanced sampling algorithms, demonstrated via simulations.

Contribution

Introduces an approximation-based data augmentation approach for models with reciprocal gamma functions, improving inference efficiency with Gibbs and Metropolis-Hastings algorithms.

Findings

Effective approximation of full conditional densities.

Enhanced Gibbs and Metropolis-Hastings algorithms.

Validated through simulation studies.

Abstract

In this paper, we introduce a new and efficient data augmentation approach to the posterior inference of the models with shape parameters when the reciprocal gamma function appears in full conditional densities. Our approach is to approximate full conditional densities of shape parameters by using Gauss's multiplication formula and Stirling's formula for the gamma function, where the approximation error can be made arbitrarily small. We use the techniques to construct efficient Gibbs and Metropolis-Hastings algorithms for a variety of models that involve the gamma distribution, Student's $t$-distribution, the Dirichlet distribution, the negative binomial distribution, and the Wishart distribution. The proposed sampling method is numerically demonstrated through simulation studies.