Rejoinder for "Probabilistic Integration: A Role in Statistical Computation?"

This article is the rejoinder for the paper "Probabilistic Integration: A Role in Statistical Computation?" to appear in Statistical Science with discussion [Briol et al., 2015]. We would first like to thank the reviewers and many of our colleagues who helped shape this paper, the editor for selecting our paper for discussion, and of course all of the discussants for their thoughtful, insightful and constructive comments. In this rejoinder, we respond to some of the points raised by the discussants and comment further on the fundamental questions underlying the paper: - Should Bayesian ideas be used in numerical analysis? Numerical analysis is concerned with the approximation of typically high or infinite-dimensional mathematical quantities using discretisations of the space on which these are defined. Different discretisation schemes lead to different numerical algorithms, whose stability and convergence properties need to be carefully assessed.