  Markov chain Monte Carlo (MCMC) allows one to generate dependent replicates
from a posterior distribution for effectively any Bayesian hierarchical model.
However, MCMC can produce a significant computational burden. This motivates us
to consider finding expressions of the posterior distribution that are
computationally straightforward to obtain independent replicates from directly.
We focus on a broad class of Bayesian latent Gaussian process (LGP) models that
allow for spatially dependent data. First, we derive a new class of
distributions we refer to as the generalized conjugate multivariate (GCM)
distribution. The GCM distribution's theoretical development is similar to that
of the CM distribution with two main differences; namely, (1) the GCM allows
for latent Gaussian process assumptions, and (2) the GCM explicitly accounts
for hyperparameters through marginalization. The development of GCM is needed
to obtain independent replicates directly from the exact posterior
distribution, which has an efficient projection/regression form. Hence, we
refer to our method as Exact Posterior Regression (EPR). Illustrative examples
are provided including simulation studies for weakly stationary spatial
processes and spatial basis function expansions. An additional analysis of
poverty incidence data from the U.S. Census Bureau's American Community Survey
(ACS) using a conditional autoregressive model is presented.
