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

André F. B. Menezes, Andrew C. Parnell, Keefe Murphy

TL;DR

This paper develops a unified finite-mixture framework for zero- and $N$-inflated count-compositional data by introducing two distributions: the zero-&-N-inflated multinomial (ZANIM) and the zero-&-N-inflated Dirichlet-multinomial (ZANIDM). Both are represented as $2^d$-component mixtures, enabling explicit sampling representations and tractable derivations of moments and marginals; ZANIM uses multinomial components while ZANIDM uses Dirichlet-multinomial components. The authors derive exact marginal distributions and moment expressions, show that the two models can support overdispersion and both positive and negative dependence, and provide Bayesian inference schemes with efficient Gibbs updates, including collapsed schemes for ZANIDM. Simulation studies demonstrate the efficiency of the proposed inference methods and illustrate the practical utility of both models for handling zero- and N-inflation in real data, with conditional improvements over existing approaches. Overall, the work offers a theoretically grounded, computationally practical toolkit for zero-/N-inflated count-compositional data and suggests avenues for extending to other count-distribution families and covariate-driven regression.

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 to illustrate the practical utility of the proposed distributions.

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

TL;DR

This paper develops a unified finite-mixture framework for zero- and -inflated count-compositional data by introducing two distributions: the zero-&-N-inflated multinomial (ZANIM) and the zero-&-N-inflated Dirichlet-multinomial (ZANIDM). Both are represented as -component mixtures, enabling explicit sampling representations and tractable derivations of moments and marginals; ZANIM uses multinomial components while ZANIDM uses Dirichlet-multinomial components. The authors derive exact marginal distributions and moment expressions, show that the two models can support overdispersion and both positive and negative dependence, and provide Bayesian inference schemes with efficient Gibbs updates, including collapsed schemes for ZANIDM. Simulation studies demonstrate the efficiency of the proposed inference methods and illustrate the practical utility of both models for handling zero- and N-inflation in real data, with conditional improvements over existing approaches. Overall, the work offers a theoretically grounded, computationally practical toolkit for zero-/N-inflated count-compositional data and suggests avenues for extending to other count-distribution families and covariate-driven regression.

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 to illustrate the practical utility of the proposed distributions.
Paper Structure (22 sections, 9 theorems, 66 equations, 3 figures, 8 tables, 2 algorithms)

This paper contains 22 sections, 9 theorems, 66 equations, 3 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Derivation of the distributions
  4. Zero-&-N-inflated multinomial distribution
  5. Zero-&-N-inflated Dirichlet-multinomial distribution
  6. Properties of ZANIM and ZANIDM
  7. Marginal distribution of Yj
  8. Moments
  9. Bayesian inference for ZANIM and ZANIDM
  10. ZANIM
  11. ZANIDM
  12. Simulation studies
  13. Comparison of MCMC algorithms for ZANIDM
  14. Simulated data analysis examples
  15. Discussion
  16. ...and 7 more sections

Key Result

Proposition 2.1

The zero-&-$N$-inflated multinomial distribution, denoted by $\mathbf{Y} \sim \operatorname{ZANIM}_d\lbrack N, \bm{\theta}, \bm{\zeta} \rbrack$, is a finite mixture distribution of $2^d$ components with PMF given by where $\bm{\theta} = (\theta_1, \ldots, \theta_d)$, with $\theta_j \geq 0$ and $\sum_{j=1}^d\theta_j=1$, $\theta_j^{\mathcal{K}} = \theta_j/(1 - \sum_{\ell \in \mathcal{K}}\,\theta_{\

Figures (3)

  • Figure 1: Marginal PMFs of ZANIM (red circles) and ZANIDM (blue triangles) with respective parameters $\bm{\theta} = (0.05, 0.70, 0.25)$ for ZANIM and $\bm{\alpha} = (2.0, 28.0, 10.0)$ for ZANIDM, along with $\bm{\zeta} = (0.05, 0.15, 0.10)$, and $N=30$ trials in each case.
  • Figure 2: Comparison of efficiency and parameter recovery of the $\bm{\alpha}$ (left) and $\bm{\zeta}$ (right) parameters for different ZANIDM inference schemes. All metrics are averaged over the $d=20$ categories and $R=50$ replicates simulated from ZANIDM with $N = 200$ trials and varying sample size $\{50, 200, 500\}$. A: effective sample size ratio; B: overall relative bias based on the posterior mean; C: overall coverage probability of the $95\%$ credible interval.
  • Figure 3: Empirical relative frequency estimates (grey bars) of the observed categories $y_j$, with the corresponding model estimates (where available) from the posterior predictive distributions of ZANIM (red circles) and ZANIDM (blue triangles). The points represent the means and the error-bars represent the corresponding $95\%$ prediction intervals. A: DGP from ZANIM. B: DGP from ZANIDM.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 21 more