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 Hölder 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 Hölder 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.

中文翻译：

---

用于一致重建内含物的贝叶斯方法

本文考虑使用两种已知且流行的前推先验分布（星形先验分布和水平集先验分布）在非线性反问题中进行包含检测的贝叶斯方法。我们分析了小测量噪声限制下相应后验分布的收敛性。方法论是通用的；它适用于由高斯随机场的任何霍尔德连续变换产生的先验，并且适用于一系列反问题。水平集和星形先验分布是 Hölder 连续变换下的前推先验的示例，它们利用了包含检测问题的结构。我们证明了相应的后验均值在适当的概率意义上收敛到基本事实。对二维定量光声断层扫描问题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夹杂物的能力。