# Optimality Criteria for Probabilistic Numerical Methods

## TL;DR

This paper explores an optimality criterion from Bayesian experimental design for probabilistic numerical methods, highlighting differences from average-case approaches and proposing regimes for developing optimal methods.

## Contribution

It introduces a novel optimality criterion for probabilistic numerical methods based on Bayesian experimental design, contrasting it with traditional average-case analysis.

## Key findings

- Optimal information differs from average-case optimal information.
- Different regimes for developing optimal probabilistic numerical methods.
- Connection established between Bayesian experimental design and numerical analysis.

## Abstract

It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that standard approaches from the decision-theoretic framework are neither appropriate nor sufficient. Instead, we consider a particular optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an average-case-optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes in which optimal probabilistic numerical methods can be developed.