Likelihood Ratios and Inference for Poisson Channels

In recent years, infinite-dimensional methods have been introduced for Gaussian channel estimation. The aim of this paper is to study the application of similar methods to Poisson channels. In particular, we compute the noncausal conditional mean 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 a new proof of a very deep and remarkable formula in Information Theory obtained recently in the literature and relating the derivatives of the input-output mutual information of a general Poisson channel and the conditional mean estimator of the input regardless the distribution of the latter. The use of the aforementioned stochastic analysis techniques allows us to extend these results to more general channels such as mixed Gaussian-Poisson channels.