Adaptive sampling for Bayesian geospatial models

Bayesian hierarchical modeling with Gaussian process random effects provides a popular approach for analyzing point-referenced spatial data. For large spatial data sets, however, generic posterior sampling is infeasible due to the extremely high computational burden in decomposing the spatial correlation matrix. In this paper, we propose an efficient algorithm-the adaptive griddy Gibbs (AGG) algorithm-to address the computational issues with large spatial data sets. The proposed algorithm dramatically reduces the computational complexity. We show theoretically that the proposed method can approximate the real posterior distribution accurately. The sufficient number of grid points for a required accuracy has also been derived. We compare the performance of AGG with that of the state-of-the-art methods in simulation studies. Finally, we apply AGG to spatially indexed data concerning building energy consumption.