Integration for and as Bayesian inference

Abstract

Integration is a key operation for probabilistic inference. In the first part of the talk, we will look at a very specific, yet ubiquitous and surprisingly hard integration problem: integrals over _linearly_ constrained multivariate Gaussian densities. One key observation is that elliptical slice sampling (ESS) allows to draw rejection-free samples from such domains. Together with multilevel splitting methods, the logarithm of the integral value can be efficiently computed by decomposing the integral into easier-to-compute conditional probabilities. The second part will focus on integration from a perspective of probabilistic numerics. Bayesian quadrature (BQ) treats numerical integration as an inference problem by constructing posterior measures over integrals given observations, i.e. evaluations of the integrand. Besides providing sound uncertainty estimates, the probabilistic approach permits the inclusion of prior knowledge about properties of the integrand and leverages active learning schemes for node selection as well as transfer learning schemes, e.g. when multiple similar integrals have to be jointly estimated.

Notes

• Alexandra Gessner is a Ph.D. candidate at the Max Planck Institute for Intelligent Systems in Tübingen. Her website can be found here.