  In this article, we consider computing expectations w.r.t. probability
measures which are subject to discretization error. Examples include partially
observed diffusion processes or inverse problems, where one may have to
discretize time and/or space, in order to practically work with the probability
of interest. Given access only to these discretizations, we consider the
construction of unbiased Monte Carlo estimators of expectations w.r.t. such
target probability distributions. It is shown how to obtain such estimators
using a novel adaptation of randomization schemes and Markov simulation
methods. Under appropriate assumptions, these estimators possess finite
variance and finite expected cost. There are two important consequences of this
approach: (i) unbiased inference is achieved at the canonical complexity rate,
and (ii) the resulting estimators can be generated independently, thereby
allowing strong scaling to arbitrarily many parallel processors. Several
algorithms are presented, and applied to some examples of Bayesian inference
problems, with both simulated and real observed data.
