Bayesian probabilistic numerical methods are a set of tools providing
posterior distributions on the output of numerical methods. The use of these
methods is usually motivated by the fact that they can represent our
uncertainty due to incomplete/finite information about the continuous
mathematical problem being approximated. In this paper, we demonstrate that
this paradigm can provide additional advantages, such as the possibility of
transferring information between several numerical methods. This allows users
to represent uncertainty in a more faithful manner and, as a by-product,
provide increased numerical efficiency. We propose the first such numerical
method by extending the well-known Bayesian quadrature algorithm to the case
where we are interested in computing the integral of several related functions.
We then prove convergence rates for the method in the well-specified and
misspecified cases, and demonstrate its efficiency in the context of
multi-fidelity models for complex engineering systems and a problem of global
illumination in computer graphics.