  Conventional Monte Carlo simulations are stochastic in the sense that the
acceptance of a trial move is decided by comparing a computed acceptance
probability with a random number, uniformly distributed between 0 and 1. Here
we consider the case that the weight determining the acceptance probability
itself is fluctuating. This situation is common in many numerical studies. We
show that it is possible to construct a rigorous Monte Carlo algorithm that
visits points in state space with a probability proportional to their average
weight. The same approach has the potential to transform the methodology of a
certain class of high-throughput experiments or the analysis of noisy datasets.
