Modeling Sums of Exchangeable Binary Variables

TL;DR

This paper introduces a new probabilistic model for sums of exchangeable binary variables, leveraging completely monotone functions and gamma distribution Laplace transforms, and demonstrates its advantages over existing models through Monte Carlo simulations and data applications.

Contribution

It presents a novel approximation model for exchangeable binary sums, improving estimation of success probabilities and correlations over the beta binomial model.

Findings

The new model outperforms beta binomial in estimating success probabilities.

It provides more accurate correlation estimates between variables.

Application to real data sets confirms its practical utility.

Abstract

We introduce a new model for sums of exchangeable binary random variables. The proposed distribution is an approximation to the exact distributional form, and relies on the theory of completely monotone functions and the Laplace transform of a gamma distribution function. Using Monte Carlo methods, we show that this new model compares favorably to the beta binomial model with respect to estimating the success probability of the Bernoulli trials and the correlation between any two variables in the exchangeable set. We apply the new methodology to two classic data sets and the results are summarized.