  A research frontier has emerged in scientific computation, wherein numerical
error is regarded as a source of epistemic uncertainty that can be modelled.
This raises several statistical challenges, including the design of statistical
methods that enable the coherent propagation of probabilities through a
(possibly deterministic) computational work-flow. This paper examines the case
for probabilistic numerical methods in routine statistical computation. Our
focus is on numerical integration, where a probabilistic integrator is equipped
with a full distribution over its output that reflects the presence of an
unknown numerical error. Our main technical contribution is to establish, for
the first time, rates of posterior contraction for these methods. These show
that probabilistic integrators can in principle enjoy the "best of both
worlds", leveraging the sampling efficiency of Monte Carlo methods whilst
providing a principled route to assess the impact of numerical error on
scientific conclusions. Several substantial applications are provided for
illustration and critical evaluation, including examples from statistical
modelling, computer graphics and a computer model for an oil reservoir.
