Bayesian Hierarchical Models With Conjugate Full-Conditional Distributions for Dependent Data From the Natural Exponential Family

摘要

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 something wecall the "conjugate multivariate distribution," which is motivated by theunivariate distribution introduced in Diaconis and Ylvisaker (1979).Furthermore, we provide substantial theoretical and methodological developmentincluding: results regarding conditional distributions, an asymptoticrelationship with the multivariate normal distribution, conjugate priordistributions, and full-conditional distributions for a Gibbs sampler. Theresults in this manuscript are extremely general, and can be adapted to manydifferent settings. We demonstrate the proposed methodology through simulatedexamples and analyses based on estimates obtained from the US Census Bureaus'American Community Survey (ACS).