Bayesian Hierarchical Models for Count Data

This dissertation focuses on the development of methodology for the analysis of multivariate count responses. Such contexts present a number of unique modeling challenges that are not well handled by standard models for count data which have restrictive mean-variance and correlation structures. In addition to being high-dimensional, sparse and overdispersed, multivariate count data often exhibits complicated dependencies across categories and samples that must be accounted for in order to obtain accurate inference. Three Bayesian modeling strategies are presented to handle these challenges and produce accurate, interpretable inference with uncertainty quantification. The first model incorporates novel nonlocal priors for variable selection which outperform existing alternatives, and introduces a process convolutions sub-model to handle temporally dependent responses taken over uneven sampling intervals. The second applies Bayesian nonparametric (BNP) methods based on a dependent Dirichlet process mixture to flexibly model how category abundance levels and zero inflation are related to covariates. The BNP approach facilitates community level comparisons across experimental conditions through density estimates that provide additional insights over simple statistical tests or ordination analysis. The third model employs a directed acyclic graph (DAG) to identify related response categories. The graphical model does a better job uncovering network relationships than alternatives based on simple marginal correlations, and, unlike simpler count data models, handles cross-category dependence in a principled manner