  We present Bayesian techniques for solving inverse problems which involve
mean-square convergent random approximations of the forward map. Noisy
approximations of the forward map arise in several fields, such as multiscale
problems and probabilistic numerical methods. In these fields, a random
approximation can enhance the quality or the efficiency of the inference
procedure, but entails additional theoretical and computational difficulties
due to the randomness of the forward map. A standard technique to address this
issue is to combine Monte Carlo averaging with Markov chain Monte Carlo
samplers, as for example in the pseudo-marginal Metropolis--Hastings methods.
In this paper, we consider mean-square convergent random approximations, and
quantify how Monte Carlo errors propagate from the forward map to the solution
of the inverse problems. Moreover, we review and describe simple techniques to
solve such inverse problems, and compare performances with a series of
numerical experiments.
