Summary The paper presents a method for assessing the uncertainty in the evaluation of an expectation over the output of a complex simulation model given uncertainty in the model parameters. Such simulation models take a long time to solve, given a set of parameters, so the task of averaging over the outputs of the simulation given uncertainty in the parameters is challenging. One cannot simply run the model so many times that error in the estimate of the integral is controlled. The authors approach the problem as an inference task. Given samples from the parameter posterior one must infer the posterior over the integral of interest.

This paper studies the numerical computation of integrals, representing estimates or predictions, over the output $f(x)$ of a computational model with respect to a distribution $p(\mathrm{d}x)$ over uncertain inputs $x$ to the model. For the functional cardiac models that motivate this work, neither $f$ nor $p$ possess a closed-form expression and evaluation of either requires $\approx$ 100 CPU hours, precluding standard numerical integration methods. Our proposal is to treat integration as an estimation problem, with a joint model for both the a priori unknown function $f$ and the a priori unknown distribution $p$. The result is a posterior distribution over the integral that explicitly accounts for dual sources of numerical approximation error due to a severely limited computational budget. This construction is applied to account, in a statistically principled manner, for the impact of numerical errors that (at present) are confounding factors in functional cardiac model assessment.