Low-Cost Bayesian Inference for Additive Approximate Gaussian Process
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Gaussian process models have been widely used in spatial/spatio-temporal statistics and uncertainty quantification. The use of a separable covariance function in these cases has computational advantages, but it ignores the potential interaction among space/time and input/output variables. In this paper we propose an additive covariance function approximation to allow for the interaction among variables in the resulting Gaussian process. The proposed method approximates the covariance function through two distinct components: a computational-complexity-reduction covariance function and a separable covariance function. The first component captures the large scale non-separable variation while the second component captures the separable variation. As shown in this paper, the new covariance structure can greatly improve the predictive accuracy but loses the computational advantages of its two distinct components. To alleviate this computational difficulty, we develop a fully conditional Markov chain Monte Carlo algorithm based on the hierarchical representation of the model. The computational cost of this algorithm is equivalent to the additive cost from the two components. The computational and inferential benefits of this new additive approximation approach are demonstrated with synthetic examples and Eastern U.S. ozone data.
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