The Vector Poisson Channel: On the Linearity of the Conditional Mean Estimator
This work addresses theoretical properties of estimators in Poisson noise for researchers in statistics and signal processing, but it is incremental as it builds on known Gaussian results.
The paper investigates conditions under which the conditional mean estimator in vector Poisson noise becomes linear, finding that only a product gamma prior distribution induces linearity and that non-zero dark current prevents linearity. It also shows that near-linearity in mean squared error implies the prior's characteristic function is close to that of a product gamma distribution.
This work studies properties of the conditional mean estimator in vector Poisson noise. The main emphasis is to study conditions on prior distributions that induce linearity of the conditional mean estimator. The paper consists of two main results. The first result shows that the only distribution that induces the linearity of the conditional mean estimator is a product gamma distribution. Moreover, it is shown that the conditional mean estimator cannot be linear when the dark current parameter of the Poisson noise is non-zero. The second result produces a quantitative refinement of the first result. Specifically, it is shown that if the conditional mean estimator is close to linear in a mean squared error sense, then the prior distribution must be close to a product gamma distribution in terms of their characteristic functions. Finally, the results are compared to their Gaussian counterparts.