# Bayesian Probabilistic Numerical Methods

## TL;DR

This paper formalizes Bayesian probabilistic numerical methods as solutions to inverse problems, providing conditions for their well-definition, convergence, and compositional use in complex tasks, bridging numerical analysis and uncertainty quantification.

## Contribution

It establishes a rigorous Bayesian framework for probabilistic numerics, including convergence analysis and methods for composing solutions to complex numerical problems.

## Key findings

- Bayesian probabilistic numerical methods are well-defined under general conditions.
- A numerical approximation scheme with proven asymptotic convergence is proposed.
- The framework is extended to pipelines of computation for complex tasks.

## Abstract

The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with a challenging industrial application presented.