dorsal/arxiv
View SchemaA scale invariant Bayesian method to solve linear inverse problems
| Authors | A. Mohammad-Djafari, Jérôme Idier |
|---|---|
| Categories | |
| ArXiv ID | physics/0111125 |
| URL | https://arxiv.org/abs/physics/0111125 |
Abstract
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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"abstract": "In this paper we propose a new Bayesian estimation method to solve linear\ninverse problems in signal and image restoration and reconstruction problems\nwhich has the property to be scale invariant. In general, Bayesian estimators\nare {\\em nonlinear} functions of the observed data. The only exception is the\nGaussian case. When dealing with linear inverse problems the linearity is\nsometimes a too strong property, while {\\em scale invariance} often remains a\ndesirable property. As everybody knows one of the main difficulties with using\nthe Bayesian approach in real applications is the assignment of the direct\n(prior) probability laws before applying the Bayes\u0027 rule. We discuss here how\nto choose prior laws to obtain scale invariant Bayesian estimators. In this\npaper we discuss and propose a familly of generalized exponential probability\ndistributions functions for the direct probabilities (the prior $p(\\xb)$ and\nthe likelihood $p(\\yb|\\xb)$), for which the posterior $p(\\xb|\\yb)$, and,\nconsequently, the main posterior estimators are scale invariant. Among many\nproperties, generalized exponential can be considered as the maximum entropy\nprobability distributions subject to the knowledge of a finite set of\nexpectation values of some knwon functions.",
"arxiv_id": "physics/0111125",
"authors": [
"A. Mohammad-Djafari",
"J\u00e9r\u00f4me Idier"
],
"categories": [
"physics.data-an"
],
"title": "A scale invariant Bayesian method to solve linear inverse problems",
"url": "https://arxiv.org/abs/physics/0111125"
},
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