This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any H\"older continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under H\"older continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.

翻译：本文考虑在非线性反问题中采用贝叶斯方法进行包含物检测，使用两种已知且流行的推前先验分布：星形先验分布和水平集先验分布。我们分析了在小测量噪声极限下相应后验分布的收敛性。该方法具有通用性；它适用于任何由高斯随机场的Hölder连续变换生成的先验分布，并可用于一系列反问题。水平集先验分布和星形先验分布是Hölder连续变换下推前先验的示例，这些变换利用了包含物检测问题的结构。我们证明了相应的后验均值在适当的概率意义下收敛到真实值。在二维定量光声层析成像问题上的数值测试展示了该方法。结果突出了后验分布的收敛特性以及该方法检测具有足够规则边界的包含物的能力。