Probabilistic Integration: A Role in Statistical Computation?

Abstract

A research frontier has emerged in scientific computation, wherein discretisation error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow, in order to assess the impact of discretisation error on the computer output. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the fact that the integrand has been discretised. Our main technical contribution is to establish, for the first time, rates of posterior contraction for one such method. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir. arXiv:1512.00933v6 [stat.ML] 18 Oct 2017 2 computer code to reproduce experiments reported in this paper can be downloaded from http://www. warwick.ac.uk/fxbriol/probabilistic_integration.

A modern retrospective on probabilistic numerics

“…A notable feature of PN research since 2010 is the way that it has advanced on a broad front. The topic of quadrature/cubature, in the tradition of Sul din and Larkin, continues to be well represented: see, e.g., [Briol et al, 2019, Gunter et al, 2014, Karvonen et al, 2018b, Osborne et al, 2012a,b, Särkkä et al, 2016, Xi et al, 2018 and [Ehler et al, 2019, Jagadeeswaran and Hickernell, 2018, Karvonen et al, 2018a, 2019. The Bayesian approach to global optimisation continues to be widely used [Chen et al, 2018, Snoek et al, 2012, whilst probabilistic perspectives on quasi-Newton methods [Hennig and Kiefel, 2013] and line search methods [Mahsereci and Hennig, 2015] have been put forward.…”

Symmetry exploits for Bayesian cubature methods

“…Evaluation of L i (ω ω ω i ) involves a call to an environment map (in this case, a picture of a lake in California; see [8]), which is associated with a computational communication cost. The illumination integral must be computed for each of the red, green, and blue (RGB) colour channels; we treat the integration problems corresponding to different colour channels as statistically independent.…”

“…as N → ∞ has been studied in both the well-specified [4,60,7,19,8] and mis-specified [28,29] regimes. Some relationships between the posterior mean estimator and classical cubature methods have been documented in [16,56,30].…”

Fast automatic Bayesian cubature using lattice sampling

“…(3) Rather than relying on strong assumptions about the integrand, such as an upper bound on its variance or total variation, we construct a stopping criterion that is based on a credible interval arising from a Bayesian approach to the problem. We build upon the work of Briol et al [1], Diaconis [5], O'Hagan [18], Ritter [22], Rasmussen and Ghahramani [20], and others. Our algorithm is an example of probabilistic numerics.…”