\begin{abstract}
We propose a novel approach for estimating conditional expectations through the framework of probabilistic numerical methods (such as Bayesian quadrature) which, unlike competitors such as least-squares Monte Carlo, allows us to incorporate prior information about the integrands. As a result, our method 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.
\end{abstract}


 % Conditional expectations are of primary interest in many machine learning applications. 
    % Unfortunately, current approaches to estimate conditional expectations do not fully exploit either the structure of the space of conditioning random variables or the space of conditioned random variables. 
    % From a Bayesian viewpoint, we demonstrate that leveraging appropriate prior information on the structure of both two spaces can improve the convergence rate and provide uncertainty quantification.
    % We propose a two-stage Gaussian process approach that can systematically specify prior beliefs about these two spaces in the form of covariance functions and we show a wide array of empirical evidence to support the claimed properties of our approach.