解线性逆问题的尺度不变贝叶斯方法
A scale invariant Bayesian method to solve linear inverse problems
摘要：
本文提出了一种新的贝叶斯估计方法来解决信号和图像恢复与重建问题中的线性逆问题，该方法具有尺度不变的性质。一般来说，贝叶斯估计量是观测数据的{ \em非线性}函数。唯一的例外是高斯的情况。当处理线性逆问题时，线性有时是一个太强的属性，而{ \em尺度不变性}通常仍然是一个理想的属性。众所周知，在实际应用中使用贝叶斯方法的主要困难之一是在应用贝叶斯规则之前分配直接（先验）概率定律。我们在这里讨论如何选择先验律来获得尺度不变的贝叶斯估计。本文讨论并提出了一类直接概率（先验$p(\xb)$和似然$p(\yb|\xb)$）的广义指数概率分布函数，它的后验$p(\xb|\yb)$和主要的后验估计是尺度不变的。在众多的性质中，广义指数可以被认为是最大熵概率分布的知识有限的期望值的一些已知的功能。
英文摘要：
In this paper we propose a new Bayesian estimation method to solve linear inverse problems in signal and image restoration and reconstruction problems which has the property to be scale invariant. In general, Bayesian estimators are {\em nonlinear} functions of the observed data. The only exception is the Gaussian case. When dealing with linear inverse problems the linearity is sometimes a too strong property, while {\em scale invariance} often remains a desirable property. As everybody knows one of the main difficulties with using the Bayesian approach in real applications is the assignment of the direct (prior) probability laws before applying the Bayes' rule. We discuss here how to choose prior laws to obtain scale invariant Bayesian estimators. In this paper we discuss and propose a familly of generalized exponential probability distributions functions for the direct probabilities (the prior $p(\xb)$ and the likelihood $p(\yb|\xb)$), for which the posterior $p(\xb|\yb)$, and, consequently, the main posterior estimators are scale invariant. Among many properties, generalized exponential can be considered as the maximum entropy probability distributions subject to the knowledge of a finite set of expectation values of some knwon functions.
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