Bayesian probabilistic numerical methods

Abstract

The increasing complexity of computer models used to solve contemporary inference problems has been set against a decreasing rate of improvement in processor speed in recent years. As a result, in many of these problems numerical error is a challenge for practitioners. However, while there has been a recent push towards rigorous quantification of uncertainty in inference problems based upon computer models, numerical error is still largely required to be driven down to a level at which its impact on inferences is negligible. Probabilistic numerical methods have been proposed to alleviate this; these are a class of numerical methods that return probabilistic uncertainty quantification for their numerical error. The attraction of such methods is clear: if numerical error in the computer model and uncertainty in an inference problem are quantified in a unified framework then careful tuning of numerical methods to mitigate the impact of numerical error on inferences could become unnecessary. In this thesis we introduce the class of Bayesian probabilistic numerical methods, whose uncertainty has a strict and rigorous Bayesian interpretation. A number of examples of conjugate Bayesian probabilistic numerical methods are presented before we present analysis and algorithms for the general case, in which the posterior distribution does not posess a closed form. We conclude by studying how these methods can be rigorously composed to yield Bayesian pipelines of computation. Throughout we present applications of the developed methods to real-world inference problems, and indicate that the uncertainty quantification provided by these methods can be of significant practical use