  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.
