反问题的全贝叶斯方法
A full Bayesian approach for inverse problems
摘要：
本文的主要目的是介绍贝叶斯推理的一些一般概念，更具体地说，反问题中的超参数估计。我们考虑一个一般的线性情况下，我们给出了一些数据$\yb$有关的未知参数$\xb$的$\yb=\Ab \xb+\nb$和我们可以分配的概率定律$p(\xb|\thetab)$，$p(\yb|\xb,\betab)$，$p(\betab)$和$p(\thetab)$。然后主要讨论如何单独地或它们的任意组合地推断$\xb$、$\thetab$和$\betab$。不同的情况进行了讨论和讨论。作为一个重要的例子，我们考虑的情况下，其中$θ$和$β$是精度参数的高斯定律，我们分配伽玛先验，我们提出了一些新的和实用的算法来估计他们同时。与其他经典方法，如最大似然法的比较和链接。关键词：贝叶斯推理，超参数估计，反问题，最大似然。
英文摘要：
The main object of this paper is to present some general concepts of Bayesian inference and more specifically the estimation of the hyperparameters in inverse problems. We consider a general linear situation where we are given some data $\yb$ related to the unknown parameters $\xb$ by $\yb=\Ab \xb+\nb$ and where we can assign the probability laws $p(\xb|\thetab)$, $p(\yb|\xb,\betab)$, $p(\betab)$ and $p(\thetab)$. The main discussion is then how to infer $\xb$, $\thetab$ and $\betab$ either individually or any combinations of them. Different situations are considered and discussed. As an important example, we consider the case where $θ$ and $β$ are the precision parameters of the Gaussian laws to whom we assign Gamma priors and we propose some new and practical algorithms to estimate them simultaneously. Comparisons and links with other classical methods such as maximum likelihood are presented. Keywords: Bayesian inference, Hyperparameter estimation, Inverse problems, Maximum likelihood.
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