Likelihood ratios and Bayesian inference for Poisson channels

In recent years, infinite-dimensional methods have been introduced for the Gaussian channels estimation. The aim of this paper is to study the application of similar methods to Poisson channels. In particular we compute the Bayesian estimator of a Poisson channel using the likelihood ratio and the discrete Malliavin gradient. This algorithm is suitable for numerical implementation via the Monte-Carlo scheme. As an application we provide an new proof of the formula obtained recently by Guo, Shamai and Verdu\'u relating some derivatives of the input-output mutual information of a time-continuous Poisson channel and the conditional mean estimator of the input. These results are then extended to mixed Gaussian-Poisson channels.