Predictive risk estimation for the expectation maximization algorithm with Poisson data

In this work, we introduce a novel estimator of the predictive risk with Poisson data, when the loss function is the Kullback–Leibler divergence, in order to definearegularizationparameter’s choicerule for theexpectationmaximization (EM) algorithm. To this aim, we provea Poisson counterpartof the Stein’s Lemma for Gaussian variables, and from this result we derive the proposed estimator showing its analogies with the well-known Stein’s unbiased risk estimatorvalid for a quadraticloss. We provethat the proposedestimator is asymptotically unbiased with increasing number of measured counts, under certain mildconditionsontheregularizationmethod.We showthattheseconditionsare satisfied by the EM algorithm under the hypothesis that the underlying matrix has positive entries and then we apply this estimator to select the EM optimal reconstruction. We present some numerical tests in the case of image deconvolution, comparing the performances of the proposed estimator with other methods available in the literature, both in the inverse crime and non-inverse crime setting.