Conditional Bayesian Quadrature

TL;DR

This paper introduces a new probabilistic numerical method for efficiently estimating conditional expectations with costly evaluations, leveraging prior smoothness knowledge to improve accuracy and convergence.

Contribution

It presents a novel Bayesian quadrature-based approach that incorporates prior information about integrands, enhancing estimation efficiency and uncertainty quantification.

Findings

Provides theoretical convergence guarantees.

Demonstrates improved performance on Bayesian sensitivity analysis.

Validates effectiveness through empirical experiments.

Abstract

We propose a novel approach for estimating conditional or parametric expectations in the setting where obtaining samples or evaluating integrands is costly. Through the framework of probabilistic numerical methods (such as Bayesian quadrature), our novel approach allows to incorporates prior information about the integrands especially the prior smoothness knowledge about the integrands and the conditional expectation. As a result, our approach provides a way of quantifying uncertainty and leads to a fast convergence rate, which is confirmed both theoretically and empirically on challenging tasks in Bayesian sensitivity analysis, computational finance and decision making under uncertainty.