  We introduce a Bayesian approach for multivariate spatio-temporal prediction
for high-dimensional count-valued data. Our primary interest is when there are
possibly millions of data points referenced over different variables,
geographic regions, and times. This problem requires extensive methodological
advancements, as jointly modeling correlated data of this size leads to the
so-called "big n problem." The computational complexity of prediction in this
setting is further exacerbated by acknowledging that count-valued data are
naturally non-Gaussian. Thus, we develop a new computationally efficient
distribution theory for this setting. In particular, we introduce a
multivariate log-gamma distribution and provide substantial theoretical
development including: results regarding conditional distributions, marginal
distributions, an asymptotic relationship with the multivariate normal
distribution, and full-conditional distributions for a Gibbs sampler. To
incorporate dependence between variables, regions, and time points, a
multivariate spatio-temporal mixed effects model (MSTM) is used. The results in
this manuscript are extremely general, and can be used for data that exhibit
fewer sources of dependency than what we consider (e.g., multivariate,
spatial-only, or spatio-temporal-only data). Hence, the implications of our
modeling framework may have a large impact on the general problem of jointly
modeling correlated count-valued data. We show the effectiveness of our
approach through a simulation study. Additionally, we demonstrate our proposed
methodology with an important application analyzing data obtained from the
Longitudinal Employer-Household Dynamics (LEHD) program, which is administered
by the U.S. Census Bureau.
