Maximum a posteriori testing in statistical inverse problems
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915209934798848 |
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| author | Kretschmann, Remo Werner, Frank |
| author_facet | Kretschmann, Remo Werner, Frank |
| contents | 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 $\langleφ, u^\dagger\rangle$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximum a posteriori testing in statistical inverse problems Kretschmann, Remo Werner, Frank Statistics Theory Numerical Analysis 47A52, 62F15, 62G10, 62G20, 65J20, 65J22 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 $\langleφ, u^\dagger\rangle$ 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. |
| title | Maximum a posteriori testing in statistical inverse problems |
| topic | Statistics Theory Numerical Analysis 47A52, 62F15, 62G10, 62G20, 65J20, 65J22 |
| url | https://arxiv.org/abs/2402.00686 |