Varying-coefficient models with isotropic Gaussian process priors

We study learning problems in which the conditional distribution of the output given the input varies as a function of additional task variables. In varying-coefficient models with Gaussian process priors, a Gaussian process generates the functional relationship between the task variables and the parameters of this conditional. Varying-coefficient models subsume multitask models—such as hierarchical Bayesian models—but also generalizations in which the conditional varies continuously, for instance, in time or space. However, Bayesian inference in varying-coefficient models is generally intractable. We show that for varying-coefficient models with isotropic Gaussian process priors, inference can be carried out efficiently in the dual. We observe that dual inference resolves to multitask learning using task and instance kernels, and that inference for hierarchical Bayesian models can be carried out efficiently in the dual using graph-Laplacian kernels. We report on experiments for geospatial prediction problems.