\section{Conclusion}

In this paper, 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 that CBQ exhibits an accelerated 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. 
