线性逆问题贝叶斯反演的标度不变马尔可夫模型
Scale Invariant Markov Models for Bayesian Inversion of Linear Inverse Problems
摘要：
在求解线性逆问题的贝叶斯方法中，需要指定用于计算后验律的先验律。也可以定义成本函数，以便为依赖于数据和超参数的各种贝叶斯估计器提供通用工具。高斯的情况下除外，这些估计不是线性的，所以依赖于测量的规模。本文将比线性更弱的性质，即尺度不变性质（SIP）加在贝叶斯估计上，首先给出了线性估计的一些结果，然后引入并证明了尺度不变公理。我们表明，任意选择的尺度测量可以避免，如果估计有这个SIP。经典的正则化程序的一些例子被证明是尺度不变的。然后，我们调查类贝叶斯估计，满足此SIP的一般条件，以及其后果的成本函数和先验定律。我们还表明，经典的超参数估计方法（即，极大似然和广义极大似然）可以用于超参数估计，并证明了它们的SIP性质。最后讨论了如何选择先验律来获得尺度不变的贝叶斯估计。为此，我们考虑两种情况下的先验定律：{ \em熵先验定律}和{ \em一阶马尔可夫模型}。在相关的先前作品[Mohammad-Djafari 90，Mohammad-Djafari 93]中，SIP约束已经被研究用于熵先验定律的情况。在本文中，扩展到一阶马尔可夫模型的情况下提供。关键词：贝叶斯估计，尺度不变性，马尔可夫模型，逆问题，图像重建，先验模型选择
英文摘要：
In a Bayesian approach for solving linear inverse problems one needs to specify the prior laws for calculation of the posterior law. A cost function can also be defined in order to have a common tool for various Bayesian estimators which depend on the data and the hyperparameters. The Gaussian case excepted, these estimators are not linear and so depend on the scale of the measurements. In this paper a weaker property than linearity is imposed on the Bayesian estimator, namely the scale invariance property (SIP).First, we state some results on linear estimation and then we introduce and justify a scale invariance axiom. We show that arbitrary choice of scale measurement can be avoided if the estimator has this SIP. Some examples of classical regularization procedures are shown to be scale invariant. Then we investigate general conditions on classes of Bayesian estimators which satisfy this SIP, as well as their consequences on the cost function and prior laws. We also show that classical methods for hyperparameters estimation (i.e., Maximum Likelihood and Generalized Maximum Likelihood) can be introduced for hyperparameters estimation, and we verify the SIP property for them. Finally we discuss how to choose the prior laws to obtain scale invariant Bayesian estimators. For this, we consider two cases of prior laws: {\em entropic prior laws} and {\em first-order Markov models}. In related preceding works [Mohammad-Djafari90,Mohammad-Djafari93], the SIP constraints have been studied for the case of entropic prior laws. In this paper extension to the case of first-order Markov models is provided. KEYWORDS: Bayesian estimation, Scale invariance, Markov modelling, Inverse Problems, Image reconstruction, Prior model selection
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