Maximum a posteriori testing in statistical inverse problems
Remo Kretschmann, Frank Werner
TL;DR
The paper develops a MAP-based Bayesian testing framework for local features in linear inverse problems with Gaussian priors, establishing that the MAP test is a regularized hypothesis test with explicit probe $\Phi_{\mathrm MAP}$. It provides both a priori and a posteriori strategies for prior covariance, deriving power bounds under spectral source conditions and analyzing asymptotic behavior as noise $\sigma$ decreases. The approach is validated through numerical simulations on deconvolution, differentiation, and backward heat problems, showing strong, often superior, finite-sample power and controlled size, even in severely ill-posed settings. The work advances local feature inference in Bayesian inverse problems, offering practical, data-driven testing procedures and paving the way for broader priors and feature choices in future research.
Abstract
This paper is concerned with a Bayesian approach to testing hypotheses in statistical inverse problems. Based on the posterior distribution $Π\left(\cdot |Y = y\right)$, 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.