In this work, we consider the inverse scattering problem of determining an unknown refractive index from the far-field measurements using the nonparametric Bayesian approach. We use a collection of large ‘samples’, which are noisy discrete measurements taking from the scattering amplitude. We will study the frequentist property of the posterior distribution as the sample size tends to infinity. Our aim is to establish the consistency of the posterior distribution with an explicit contraction rate in terms of the sample size. We will consider two different priors on the space of parameters. The proof relies on the stability estimates of the forward and inverse problems. Due to the ill-posedness of the inverse scattering problem, the contraction rate is of a logarithmic type. We also show that such contraction rate is optimal in the statistical minimax sense.

中文翻译：

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逆散射问题贝叶斯方法的一致性

在这项工作中，我们考虑使用非参数贝叶斯方法从远场测量中确定未知折射率的逆散射问题。我们使用大量“样本”的集合，这些样本是从散射幅度中获取的噪声离散测量值。当样本量趋于无穷大时，我们将研究后验分布的频率特性。我们的目标是建立后验分布与样本量方面的显式收缩率的一致性。我们将考虑参数空间上的两个不同的先验。证明依赖于正向和逆向问题的稳定性估计。由于逆散射问题的不适定性，收缩率是对数类型的。我们还表明，这种收缩率在统计极小极大意义上是最优的。