Likelihood ratios and Bayesian inference for Poisson channels

TL;DR

This paper extends infinite-dimensional Bayesian estimation methods to Poisson channels, deriving new formulas for mutual information derivatives and proposing Monte Carlo algorithms for practical implementation.

Contribution

It introduces a novel approach using likelihood ratios and Malliavin gradients for Poisson channels, including extensions to mixed Gaussian-Poisson models.

Findings

Derived Bayesian estimator for Poisson channels.

Provided a new proof of mutual information derivative formula.

Extended results to Gaussian-Poisson mixed channels.

Abstract

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.