Likelihood ratios and Bayesian inference for Poisson channels

In recent years infinite-dimensional stochastic analysis methods have been introduced in the field of estimation for Gaussian channels. The aim of this note is to study the application of similar methods to Poisson channels. In particular we show that the conditional mean estimator of a Poisson channel can be expressed as a discrete logarithmic Malliavin gradient of the likelihood-ratio of the observation. The result is extended to a mixture of Brownian motion and of an independent Poisson process and then to a class of martingales with jumps and non-independent increments.