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We introduce a Bayesian approach for analyzing (possibly) high-dimensionaldependent data that are distributed according to a member from the naturalexponential family of distributions. This problem requires extensivemethodological advancements, as jointly modeling high-dimensional dependentdata leads to the so-called "big n problem." The computational complexity ofthe "big n problem" is further exacerbated when allowing for non-Gaussian datamodels, as is the case here. Thus, we develop new computationally efficientdistribution theory for this setting. In particular, we introduce the"conjugate multivariate distribution," which is motivated by the univariatedistribution introduced in Diaconis and Ylvisaker (1979). Furthermore, weprovide substantial theoretical and methodological development including:results regarding conditional distributions, an asymptotic relationship withthe multivariate normal distribution, conjugate prior distributions, andfull-conditional distributions for a Gibbs sampler. To demonstrate thewide-applicability of the proposed methodology, we provide two simulationstudies and three applications based on an epidemiology dataset, a federalstatistics dataset, and an environmental dataset, respectively.

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