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Choice of the estimator
How to automatically select the optimal hyperparameters?


We have to optimize a criterion which depends on the hyperparameters. 
It is the Maximum Likelihood Estimator on the observed image Y w.r.t.  and : 

   maximizing a probability w.r.t. and :

where  = set of images , with M grey levels
 We use Bayes law to calculate the likelihood :
 which exhibits the two following probability densities :
   probability to obtain the data Y from the initial image X,
(joint likelihood of X and hyperparameters)
  a priori probability on the image X 
calculated only taking into account the constraints
characterizes the type of regularization  -function
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André Jalobeanu - 24 Aug 1998