  In the bayesian analysis of Inverse Problems most relevant cases the forward
maps (FM, or regressor function) are defined in terms of a system of (O, P)DE's
with intractable solutions. These necessarily involve a numerical method to
find approximate versions of such solutions and lead to a numerical/approximate
posterior distribution. Recently several results have been published on the
regularity conditions required on such numerical methods to ensure converge of
the numerical to the theoretical posterior. However, more practical guidelines
are needed to ensure a suitable working numerical posterior. ]Capistran2016]
prove for ODE's that the Bayes Factor of the approximate vs the theoretical
model tends to 1 in the same order as the numerical method order. In this work
we generalize the latter paper in that we consider 1) also PDE's, 2) correlated
observations, 3) practical guidelines in a multidimensional setting and 4)
explore the use of expected Bayes Factors. This permits us to obtain bounds on
the absolute global errors to be tolerated by the FM numerical solver, which we
illustrate with some examples. Since the Bayes Factor is kept above 0.95 we
expect that the resulting numerical posterior is basically indistinguishable
from the theoretical posterior, even though we are using an approximate
numerical FM. The method is illustrated with some examples using synthetic
data.
