Predictive risk estimation for the expectation maximization algorithm with Poisson data

TL;DR: A Poisson counterpart of the Stein’s Lemma for Gaussian variables is proved, and from this result the proposed estimator is derived, showing its analogies with the well-known Stein's unbiased risk estimator valid for a quadratic loss.

Abstract: 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 define a regularization parameter's choice rule for the Expectation Maximization (EM) algorithm. To this aim, we prove a Poisson counterpart of 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 Estimator valid for a quadratic loss. We prove that the proposed estimator is asymptotically unbiased with increasing number of measured counts, under certain mild conditions on the regularization method. We show that these conditions are satisfied by the EM algorithm and then we apply this estimator to select its 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.