\section{Conclusions}
We propose CBQ, a novel algorithm which is tailored for the computation of conditional expectations in the setting where obtaining samples or evaluating functions is costly. 
We show 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 output of 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 for some helpful discussions and pointers to the literature. ZC, MN and FXB acknowledge support from the Engineering and Physical Sciences Research Council (ESPRC) through grants [EP/S021566/1] and [EP/Y022300/1].  AG was partly supported by the Gatsby Charitable Foundation.
% with grant number EP/S021566/1. 
% FXB was supported by the EPSRC grant [EP/Y022300/1].
