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 $\left\langle\varphi, u^\dagger\right\rangle$ 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：本文关注统计逆问题中假设检验的贝叶斯方法。基于后验分布 $\Pi \left(\cdot |Y = y\right)$，我们希望推断未知目标量 $u^\dagger$ 的特征 $\left\langle\varphi, u^\dagger\right\rangle$ 是否为正值。这可以通过所谓的最大后验检验实现。我们从频率学派角度分析了该检验的水平和功效等性质，并证明它符合Kretschmann等人（2024）提出的正则化检验的定义。此外，在高斯先验情况下，我们在经典谱源条件下给出了其功效的下界。数值模拟表明，该方法在中度病态和严重病态问题中均表现出优越性能。