Lognormal and Gamma Mixed Negative Binomial Regression

TL;DR

This paper introduces a novel Bayesian negative binomial regression model with efficient closed-form inference methods, enabling practical and flexible analysis of count data with random effects and prior information.

Contribution

It develops a lognormal and gamma mixed NB regression model with closed-form Bayesian inference, including Gibbs sampling and variational Bayes, for counts data analysis.

Findings

Efficient Gibbs sampling and variational Bayes algorithms are derived.

The model effectively incorporates prior information and random effects.

Algorithms are demonstrated on real datasets.

Abstract

In regression analysis of counts, a lack of simple and efficient algorithms for posterior computation has made Bayesian approaches appear unattractive and thus underdeveloped. We propose a lognormal and gamma mixed negative binomial (NB) regression model for counts, and present efficient closed-form Bayesian inference; unlike conventional Poisson models, the proposed approach has two free parameters to include two different kinds of random effects, and allows the incorporation of prior information, such as sparsity in the regression coefficients. By placing a gamma distribution prior on the NB dispersion parameter r, and connecting a lognormal distribution prior with the logit of the NB probability parameter p, efficient Gibbs sampling and variational Bayes inference are both developed. The closed-form updates are obtained by exploiting conditional conjugacy via both a compound Poisson representation and a Polya-Gamma distribution based data augmentation approach. The proposed Bayesian inference can be implemented routinely, while being easily generalizable to more complex settings involving multivariate dependence structures. The algorithms are illustrated using real examples.