  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.
