Probabilistic Integration: A Role for Statisticians in Numerical Analysis?

A research frontier has emerged in scientific computation, founded on the principle that numerical error entails epistemic uncertainty that ought to be subjected to statistical analysis. This viewpoint raises several interesting challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational pipeline. This paper examines thoroughly the case for probabilistic numerical methods in statistical computation and a specific case study is presented for Markov chain and Quasi Monte Carlo methods. A probabilistic integrator is equipped with a full distribution over its output, providing a measure of epistemic uncertainty that is shown to be statistically valid at finite computational levels, as well as in asymptotic regimes. The approach is motivated by expensive integration problems, where, as in krigging, one is willing to expend, at worst, cubic computational effort in order to gain uncertainty quantification. There, probabilistic integrators enjoy the "best of both worlds", leveraging the sampling efficiency of Monte Carlo methods whilst providing a principled route to assessment of the impact of numerical error on scientific conclusions. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and uncertainty quantification in oil reservoir modelling.