Probabilistic Integration: A Role in Statistical Computation?

TL;DR

This paper advocates for probabilistic numerical methods in statistical computation, especially numerical integration, to quantify and propagate numerical error as epistemic uncertainty, enhancing the reliability of scientific conclusions.

Contribution

It establishes the first rates of posterior contraction for probabilistic integrators, demonstrating their potential to combine sampling efficiency with principled error assessment.

Findings

Probabilistic integrators can achieve optimal posterior contraction rates.

They enable coherent propagation of numerical uncertainty in statistical workflows.

Applications include statistical modeling, computer graphics, and reservoir simulation.

Abstract

A research frontier has emerged in scientific computation, wherein numerical 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. 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 presence of an unknown numerical error. Our main technical contribution is to establish, for the first time, rates of posterior contraction for these methods. These show that probabilistic integrators can in principle enjoy the "best of both worlds", leveraging the sampling efficiency of Monte Carlo methods whilst providing a principled route to assess 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 a computer model for an oil reservoir.