This paper proposes a four-pronged approach to efficient Bayesian estimation and prediction for complex Bayesian hierarchical Gaussian models for spatial and spatiotemporal data. The method involves reparameterizing the variance/covariance structure of the model, reformulating the means structure, marginalizing the joint posterior distribution, and applying a simplex-based slice sampling algorithm. The approach permits fusion of point-source data and areal data measured at different resolutions

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... and accommodates non-spatial correlation and variance heterogeneity as well as spatial and/or temporal correlation. The method produces Markov chain Monte Carlo samplers with low autocorrelation in the output, so that fewer iterations are needed for Bayesian inference than would be the case with other sampling algorithms.