  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.
