Title: Gaussian processes in Bayesian statistics: review and some recent results

Abstract:

A nonparametric Bayesian statistical method models an infinite-dimensional unknown parameter of interest by a `prior' probability distribution and next obtains a `posterior' distribution over the unknowns using Bayes's rule. The approach is quite elegant, and popular in applied settings, for instance in inverse problems or machine learning, when it is desired to obtain not only a reconstruction (or best guess) of the unknowns, but also a quantification of the uncertainty in this reconstruction. In this talk we discuss the validity of the approach and its determinants from a non-Bayesian point of view, focusing on Gaussian processes, which are a first choice as priors for functions. We explain ways to formalise `validity', and review some theoretical results on posterior distributions resulting from such priors when used to model a regression function or density function, or a functional parameter in an inverse problem described by a differential equation. We review the role of the small ball probability of the Gaussian process as a determinant of the contraction rate of the posterior distribution, the importance of the length scale of the process, and the accuracy of credible sets for uncertainty quantification. We present recent results on approximating a posterior distribution by distributed computing, and on using linear methods to solve inverse problems resulting from some nonlinear partial differential equations.