MAP estimators for nonparametric Bayesian inverse problems in Banach\n spaces

2020/07/24 by Philipp Wacker, Wacker, Philipp
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2007.12760

openalex publication_date 2020/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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