MAP estimators for nonparametric Bayesian inverse problems in Banach spaces

TL;DR

This paper establishes the existence of MAP estimators for nonparametric Bayesian inverse problems in Banach spaces by deriving bounds on small ball probabilities, extending previous results from Hilbert spaces to Banach spaces.

Contribution

It proves the existence of MAP estimators in Banach spaces under mild conditions, filling a gap in the theoretical understanding of Bayesian inverse problems.

Findings

Derived bounds on small ball probabilities for Banach spaces.

Proved existence of MAP estimators in Banach space setting.

Extended theoretical results from Hilbert to Banach spaces.

Abstract

In order to rigorously define maximum-a-posteriori estimators for nonparametric Bayesian inverse problems for general Banach space valued parameters, we derive and prove certain previously postulated but unproven bounds on small ball probabilities. This allows us to prove existence of MAP estimators in the Banach space setting under very mild assumptions on the loglikelihood. As a similar statement so far (as far as the author is aware) only existed in the Hilbert space setting, this closes an important gap in the literature.