\begin{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.



% 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.