A Modern Retrospective on Probabilistic Numerics
C. J. Oates, T. J. Sullivan
TL;DR
The paper surveys the historical development of probabilistic numerics (PN) from early probabilistic treatments of interpolation and round-off error to modern uncertainty-quantified numerical methods. It clarifies two main PN paradigms: (i) probabilistic analysis of classical numerical methods and (ii) Bayesian probabilistic numerical methods that output distributional outputs and compose naturally in computational pipelines. It highlights key milestones such as Sul\'din and Larkin's early work, the connection between average-case analysis and Bayes rules, and the kernel-quadrature/Bayesian ODE-PDE frameworks that underpin current PN research, including adaptive and empirical Bayes approaches. The paper argues for the integration of PN into broader uncertainty quantification, probabilistic programming, and cross-disciplinary collaboration to tackle complex, high-dimensional numerical tasks with rigorous uncertainty assessment and robust decision-making.
Abstract
This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul'din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.