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Let $X$ be a non-negative random variable and let the conditionaldistribution of a random variable $Y$, given $X$, be ${Poisson}(\gamma \cdotX)$, for a parameter $\gamma \geq 0$. We identify a natural loss function suchthat: 1) The derivative of the mutual information between $X$ and $Y$ withrespect to $\gamma$ is equal to the \emph{minimum} mean loss in estimating $X$based on $Y$, regardless of the distribution of $X$. 2) When $X \sim P$ isestimated based on $Y$ by a mismatched estimator that would have minimized theexpected loss had $X \sim Q$, the integral over all values of $\gamma$ of theexcess mean loss is equal to the relative entropy between $P$ and $Q$.  For a continuous time setting where $X^T = \{X_t, 0 \leq t \leq T \}$ is anon-negative stochastic process and the conditional law of $Y^T=\{Y_t, 0\let\le T\}$, given $X^T$, is that of a non-homogeneous Poisson process withintensity function $\gamma \cdot X^T$, under the same loss function: 1) Theminimum mean loss in \emph{causal} filtering when $\gamma = \gamma_0$ is equalto the expected value of the minimum mean loss in \emph{non-causal} filtering(smoothing) achieved with a channel whose parameter $\gamma$ is uniformlydistributed between 0 and $\gamma_0$. Bridging the two quantities is the mutualinformation between $X^T$ and $Y^T$. 2) This relationship between the meanlosses in causal and non-causal filtering holds also in the case where thefilters employed are mismatched, i.e., optimized assuming a law on $X^T$ whichis not the true one. Bridging the two quantities in this case is the sum of themutual information and the relative entropy between the true and the mismatcheddistribution of $Y^T$. Thus, relative entropy quantifies the excess estimationloss due to mismatch in this setting.  These results parallel those recently found for the Gaussian channel.

Mutual Information, Relative Entropy, and Estimation in the Poisson Channel