Generating Independent Replicates Directly from the Posterior Distribution for a Class of Spatial Latent Gaussian Process Models

TL;DR

This paper introduces a new distribution called GCM for efficiently generating independent posterior replicates in spatial Gaussian process models, avoiding the computational costs of MCMC.

Contribution

We develop the GCM distribution and the EPR method to produce independent posterior samples directly from the exact distribution in spatial LGP models.

Findings

GCM enables direct sampling from complex posterior distributions.

EPR reduces computational burden compared to traditional MCMC methods.

Application to real data demonstrates practical effectiveness.

Abstract

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.