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 Laplace transforms. Finally, the results are compared to their Gaussian counterparts.

中文翻译：

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向量泊松通道：关于条件均值估计量的线性

这项工作研究了向量泊松噪声中条件均值估计器的特性。主要重点是研究导致条件均值估计量线性的先验分布条件。该论文包括两个主要结果。第一个结果表明，引起条件均值估计量线性的唯一分布是乘积伽马分布。而且，当泊松噪声的暗电流参数不为零时，条件均值估计量不能是线性的。第二个结果产生了第一个结果的定量细化。具体来说，如果条件均值估计量在均方误差意义上接近线性，则先验分布在其拉普拉斯变换方面必须接近乘积伽马分布。最后，