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.

翻译：本文研究统计逆问题中假设检验的贝叶斯方法。基于后验分布 $\Pi \left(\cdot |Y = y\right)$，我们旨在推断未知目标量 $u^\dagger$ 的特征 $\langle\varphi, u^\dagger\rangle$ 是否为正。这可通过所谓的最大后验检验实现。我们对该检验的频域性质（如检验水平和功效）进行了分析，并证明其符合 Kretschmann 等人（2024）所定义的正则化检验框架。此外，在高斯先验条件下，我们基于经典谱源条件给出了该检验功效的下界估计。数值模拟表明，该方法在中等程度和严重不适定问题中均表现出优越性能。