  Probabilistic numerics casts numerical tasks, such the numerical solution of
differential equations, as inference problems to be solved. One approach is to
model the unknown quantity of interest as a random variable, and to constrain
this variable using data generated during the course of a traditional numerical
method. However, data may be nonlinearly related to the quantity of interest,
rendering the proper conditioning of random variables difficult and limiting
the range of numerical tasks that can be addressed. Instead, this paper
proposes to construct probabilistic numerical methods based only on the final
output from a traditional method. A convergent sequence of approximations to
the quantity of interest constitute a dataset, from which the limiting quantity
of interest can be extrapolated, in a probabilistic analogue of Richardson's
deferred approach to the limit. This black box approach (1) massively expands
the range of tasks to which probabilistic numerics can be applied, (2) inherits
the features and performance of state-of-the-art numerical methods, and (3)
enables provably higher orders of convergence to be achieved. Applications are
presented for nonlinear ordinary and partial differential equations, as well as
for eigenvalue problems-a setting for which no probabilistic numerical methods
have yet been developed.
