  We generalize the results of \cite{Capistran2016} on expected Bayes factors
(BF) to control the numerical error in the posterior distribution to an
infinite dimensional setting when considering Banach functional spaces and now
in a prior setting. The main result is a bound on the absolute global error to
be tolerated by the Forward Map numerical solver, to keep the BF of the
numerical vs. the theoretical model near to 1, now in this more general
setting, possibly including a truncated, finite dimensional approximate prior
measure. In so doing we found a far more general setting to define and prove
existence of the infinite dimensional posterior distribution than that depicted
in, for example, \cite{Stuart2010}. Discretization consistency and rates of
convergence are also investigated in this general setting for the Bayesian
inverse problem.
