Computing the Dirichlet-Multinomial Log-Likelihood Function

TL;DR

This paper derives a closed-form expression for the Dirichlet-multinomial log-likelihood, enabling faster and accurate statistical inference for over-dispersed count data.

Contribution

It introduces a mathematically derived closed-form formula for the DMN log-likelihood, improving computational efficiency over existing methods.

Findings

The closed-form formula reduces computation time significantly.

The method maintains high numerical accuracy.

It facilitates faster statistical inference with DMN models.

Abstract

Dirichlet-multinomial (DMN) distribution is commonly used to model over-dispersion in count data. Precise and fast numerical computation of the DMN log-likelihood function is important for performing statistical inference using this distribution, and remains a challenge. To address this, we use mathematical properties of the gamma function to derive a closed form expression for the DMN log-likelihood function. Compared to existing methods, calculation of the closed form has a lower computational complexity, hence is much faster without comprimising computational accuracy.