\section{Conclusion}

\vspace{-2mm}
We proposed CBQ, a novel algorithm which is tailored for the computation of conditional expectations where obtaining samples or function evaluations is costly. We showed both theoretically and empirically that CBQ exhibits a fast convergence rate, and provides the additional benefit of Bayesian quantification of uncertainty. Looking forward, we believe further gains in accuracy could be obtained by developing active learning schemes to $N$, $T$, and the location of $\theta_{1:T}$ and $x_{1:N}^t$ for all $t$ in an adaptive manner. Additionally, CBQ could be extended for nested expectation problems by using a second level of BQ based on the second stage heteroscedastic GP, potentially leading to a further increase in accuracy. 



% Summary of contributions:
% - BQ algorithms for conditional expectations: advantages of a PN algorithm, able to explicitly take into account the relationship of $I$ as a function of $\theta$ through the stage two GP

% Possible future work:
% - extended algorithm for nested expectations.


% \paragraph{Ackowledgments:} The authors would like to thank Motonobu Kanagawa and Alessandro Barp for some helpful discussions and pointers to the literature. 
