Maximum a posteriori testing in statistical inverse problems

Remo Kretschmann; Frank Werner

doi:10.3934/ipi.2025015

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2025, Volume 0. Doi: 10.3934/ipi.2025015

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Maximum a posteriori testing in statistical inverse problems

Remo Kretschmann^1, and
Frank Werner^2, ,

1.
Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Straße 24–25, 14476 Potsdam, Germany
2.
Institute of Mathematics, University of Würzburg, Emil-Fischer-Straße 30, 97074 Würzburg, Germany

^*Corresponding author: Frank Werner
^*Corresponding author: Frank Werner

Received: April 2024

Revised: March 2025

Early access: April 2025

The authors are supported by the German Research Foundation (DFG) under grant WE 6204/2-1. The research of RK is partially funded by the DFG under project 318763901 — SFB 1294.

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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.

Keywords:

Mathematics Subject Classification: Primary: 47A52; Secondary: 62F15, 62G10, 62G20, 65J20, 65J22.

Citation:

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Figure 1. The function $ \varphi_{l, \beta} $ with $ l = \frac{5}{128} $ for $ \beta = 1 $ () and the corresponding $ \beta_{\mathrm{conv}}, \beta_{\mathrm{antider}} $ chosen such that $ \varphi_{l, \beta} \in \operatorname{ran} T^* $ (), and the truth $ u^\dagger $ ()

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Figure 2. Exact powers of the unregularized test (), the oracle MAP test (), and the oracle a priori MAP test (), the oracle lower bound for the power of the a priori MAP test (), the empirical power of the 2 sample MAP test (), and the empirical power () and level () of the 1 sample MAP test for the deconvolution problem with different values of $ \beta $ and $ \mu $

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Figure 3. Choice of $ \gamma_0 $ by the oracle MAP test () as well as mean (), $ 16 \% $ and $ 84 \% $ quantiles () of the choice of $ \gamma_0 $ by the a posteriori MAP test for the deconvolution problem.

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Figure 4. Exact powers of the unregularized test (), the oracle MAP test (), and the oracle a priori MAP test (), the oracle lower bound for the power of the a priori MAP test (), the empirical power of the 2 sample MAP test (), and the empirical power () and level () of the 1 sample MAP test for the differentiation problem with different values of $ \beta $ and $ \mu $

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Figure 5. Choice of $ \gamma_0 $ by the oracle MAP test () as well as mean (), $ 16 \% $ and $ 84 \% $ quantiles () of the choice of $ \gamma_0 $ by the a posteriori MAP test for the differentiation problem.

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Figure 6. Exact powers of the unregularized test (), the oracle MAP test (), and the oracle a priori MAP test (), the oracle lower bound for the power of the a priori MAP test (), the empirical power of the 2 sample MAP test (), and the empirical power () and level () of the 1 sample MAP test for the backward heat equation with $ \beta = 1 $ and different values of $ \mu $

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Figure 7. Choice of $ \gamma_0 $ by the oracle MAP test () as well as mean (), $ 16 \% $ and $ 84 \% $ quantiles () of the choice of $ \gamma_0 $ by the a posteriori MAP test for the backward heat equation.

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