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Title: Accounting for the sparsity rate in prior selection in inverse problems via generalized Student$-t$ distribution Authors:  Mircea Dumitru - CNRS - Universite Paris-Sud - CentraleSupelec (France)
Li Wang - CNRS - Universite Paris-Sud - CentraleSupelec (France) [presenting]
Ali Mohammad-Djafari - CNRS (France)
Abstract: Bayesian methods have become very common for inverse problems arising in signal and image processing. The main advantages are the possibility to propose unsupervised methods where the likelihood and prior model parameters can be estimated jointly with the main unknowns and to select prior distributions that are in accordance with the prior knowledge. In the context of sparsity enforcing priors, the selection of the prior distribution can be done also in accordance with the sparsity rate of the unknown of the model. This can be achieved based on the generalization of the Student$-t$ distribution. First, the generalization of the Student$-t$ distribution is proposed, based on its Infinite Gaussian Scaled Mixture (IGSM) model. The generalization is obtained as the marginal posterior distribution of the mean of a Gaussian distribution with unknown variance on which an a priori Inverse Gamma distribution is assigned. Then, some of its properties are discussed, namely the computation of the variance and its interpretation in the context of accounting for the sparsity rate and the links with the corresponding Inverse-Gamma distribution in the context of sparsity enforcing mechanism in the Bayesian approach. Simulations results and comparisons are presented for applications in Computed Tomography and chronobiology.