  The emergent field of probabilistic numerics has thus far lacked clear
statistical principals. This paper establishes Bayesian probabilistic numerical
methods as those which can be cast as solutions to certain inverse problems
within the Bayesian framework. This allows us to establish general conditions
under which Bayesian probabilistic numerical methods are well-defined,
encompassing both non-linear and non-Gaussian models. For general computation,
a numerical approximation scheme is proposed and its asymptotic convergence
established. The theoretical development is then extended to pipelines of
computation, wherein probabilistic numerical methods are composed to solve more
challenging numerical tasks. The contribution highlights an important research
frontier at the interface of numerical analysis and uncertainty quantification,
with a challenging industrial application presented.
