Linear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression

In this paper we shall discuss an extension to Gaussian process (GP) regression models, where the measurements are modeled as linear functionals of the underlying GP and the estimation objective is a general linear operator of the process. We shall show how this framework can be used for modeling physical processes involved in measurement of the GP and for encoding physical prior information into regression models in form of stochastic partial differential equations (SPDE). We shall also illustrate the practical applicability of the theory in a simulated application.