Posterior distribution existence and error control in Banach spaces
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We generalize the results of Christen2017 on expected Bayes factors (BF) to control the numerical error in the posterior distribution to an infinite dimensional setting when considering Banach functional spaces. 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, Stuart2010. Discretization consistency and rates of convergence are also investigated in this general setting for the Bayesian inverse problem.
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