Year: 2025
Type: article
Abstract: This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $ \Pi \left(\cdot |Y = y\right) $, we want to infer whether a feature $ \langle\varphi, u^\dagger\rangle $ of the unknown quantity of interest $ u^\dagger $ is positive. This can be done by the so-called maximum a posteriori test. We provide a frequentistic analysis of this test's properties such as level and power, and prove that it is a regularized test in the sense of Kretschmann et al. (2024). Furthermore we provide lower bounds for its power under classical spectral source conditions in case of Gaussian priors. Numerical simulations illustrate its superior performance both in moderately and severely ill-posed situations.
Source: Inverse Problems and Imaging
DOI: 10.3934/ipi.2025015
Authors: Remo Kretschmann, Frank Werner
Institutions: University of Potsdam, University of Würzburg
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4409319667