We focus upon flexible Bayesian hierarchical models for scientific data available at geo-coded locations. Investigators are increasingly turning to spatial process models to analyze such datasets. These models are customarily estimated using Markov Chain Monte Carlo (MCMC) methods, which have become especially popular for spatial modeling, given their flexibility and power to fit models that would be infeasible otherwise. However, estimating Bayesian spatial process models is undermined by prohibitive computational expenses associated with parameter estimation. Classes of low-rank spatial process models are increasingly being deployed to resolve this problem by projecting spatial effects to a lower-dimensional subspace. We discuss how a low-rank process called the ‘predictive process’ seamlessly enters the hierarchical modeling framework and helps us accrue substantial computational benefits. WIREs Comp Stat 2012, 4:59–66. doi: 10.1002/wics.187
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