Finite mixture representations of zero-and-$N$-inflated distributions for count-compositional data

TL;DR

This paper introduces a unifying finite mixture framework for zero-inflated count-compositional data, providing new theoretical characterizations and improved Bayesian inference methods, with applications to microbiome data.

Contribution

It presents a novel finite mixture representation for two multivariate zero-inflated models, including a new multinomial-based distribution, and develops efficient Bayesian inference schemes.

Findings

Finite mixture representations enable derivation of moments and marginal distributions.

Enhanced Gibbs sampling improves Bayesian inference efficiency.

Application to microbiome data demonstrates practical utility.

Abstract

We provide novel probabilistic portrayals of two multivariate models designed to handle zero-inflation in count-compositional data. We develop a new unifying framework that represents both as finite mixture distributions. One of these distributions, based on Dirichlet-multinomial components, has been studied before, but has not yet been properly characterised as a sampling distribution of the counts. The other, based on multinomial components, is a new contribution. Using our finite mixture representations enables us to derive key statistical properties, including moments, marginal distributions, and special cases for both distributions. We develop enhanced Bayesian inference schemes with efficient Gibbs sampling updates, wherever possible, for parameters and auxiliary variables, demonstrating improvements over existing methods in the literature. We conduct simulation studies to evaluate the efficiency of the Bayesian inference procedures and present applications to a human gut microbiome dataset to illustrate the practical utility of the proposed distributions.