# Bayesian Hierarchical Models with Conjugate Full-Conditional Distributions for Dependent Data from the Natural Exponential Family

## TL;DR

This paper develops a Bayesian hierarchical modeling framework for high-dimensional dependent data from the natural exponential family, introducing conjugate distributions and efficient computation methods to address the big n problem.

## Contribution

It introduces the conjugate multivariate distribution and provides theoretical and computational tools for Bayesian analysis of dependent non-Gaussian data.

## Key findings

- Efficient Gibbs sampling algorithms developed.
- Demonstrated applicability through three diverse real-world datasets.
- Established asymptotic normality relationship for the proposed models.

## Abstract

We introduce a Bayesian approach for analyzing (possibly) high-dimensional dependent data that are distributed according to a member from the natural exponential family of distributions. This problem requires extensive methodological advancements, as jointly modeling high-dimensional dependent data leads to the so-called "big n problem." The computational complexity of the "big n problem" is further exacerbated when allowing for non-Gaussian data models, as is the case here. Thus, we develop new computationally efficient distribution theory for this setting. In particular, we introduce the "conjugate multivariate distribution," which is motivated by the univariate distribution introduced in Diaconis and Ylvisaker (1979). Furthermore, we provide substantial theoretical and methodological development including: results regarding conditional distributions, an asymptotic relationship with the multivariate normal distribution, conjugate prior distributions, and full-conditional distributions for a Gibbs sampler. To demonstrate the wide-applicability of the proposed methodology, we provide two simulation studies and three applications based on an epidemiology dataset, a federal statistics dataset, and an environmental dataset, respectively.