Abstract
This special edition of Statistics and Computing is dedicated to the emerging field of Probabilistic Numerics, at the interface of statistical inference and numerical analysis, and accompanies the workshop on Probabilistic Numerics held in London, 11–13 April 2018.
As traditionally understood, numerical analysis concerns itself with the design and analysis of methods for the (approximate) solution of deterministic well-posed problems such as quadrature/cubature or the solution of a differential equation. Such a numerical method can be cast, at least conceptually, as a statistical estimator for its associated quantity of interest. Often such numerical methods are accompanied by error estimates, or are designed in such a way that a user-specified tolerance is hoped to be met; these can equally be cast, at a high conceptual level, as a kind of confidence interval. Probabilistic numerics seeks to better understand and make rigorous the use of formal statistical techniques in the numerical analysis context.
The use of statistical techniques to better characterise and understand numerical algorithms has received considerable attention in recent years. A particular success story is the widespread use of Bayesian optimisation in the training of regression and classification models in machine learning, while to a lesser extent statistical approaches to numerical cubature and the numerical solution of differential equations have been deployed in the applied context. However, there remain substantial research questions that must be addressed before the field can reach its full potential. First, these methods can involve certain modelling choices that must be user-specified (e.g. in a Bayesian context, the choice of a prior) and further research is required to assist or automate probabilistic numerical methods in this important respect. Second, until a unified statistical framework is developed that encompasses a suite of probabilistic numerical methods, it is unclear how such methods could be directly combined within the context of a complex computer simulation.