Bivariate beta distribution: parameter inference and diagnostics

TL;DR

This paper explores the bivariate beta distribution for modeling correlated proportions, focusing on parameter inference, Bayesian and frequentist estimation methods, and diagnostics for model fit through Monte Carlo experiments.

Contribution

It analyzes a specific bivariate beta construction, develops estimation techniques, and introduces diagnostics for assessing model adequacy in correlated proportion data.

Findings

Bayesian and frequentist estimators perform well under various settings.

Diagnostics effectively identify model misspecification.

The proposed methods are practical for real-world correlated proportion analysis.

Abstract

Correlated proportions appear in many real-world applications and present a unique challenge in terms of finding an appropriate probabilistic model due to their constrained nature. The bivariate beta is a natural extension of the well-known beta distribution to the space of correlated quantities on $[0, 1]^2$. Its construction is not unique, however. Over the years, many bivariate beta distributions have been proposed, ranging from three to eight or more parameters, and for which the joint density and distribution moments vary in terms of mathematical tractability. In this paper, we investigate the construction proposed by Olkin & Trikalinos (2015), which strikes a balance between parameter-richness and tractability. We provide classical (frequentist) and Bayesian approaches to estimation in the form of method-of-moments and latent variable/data augmentation coupled with Hamiltonian Monte Carlo, respectively. The elicitation of bivariate beta as a prior distribution is also discussed. The development of diagnostics for checking model fit and adequacy is explored in depth with the aid of Monte Carlo experiments under both well-specified and misspecified data-generating settings. Keywords: Bayesian estimation; bivariate beta; correlated proportions; diagnostics; method of moments.