Mutual Information of a class of Poisson-type Channels using Markov Renewal Theory
The mutual information (MI) of Poisson-type channels has been linked to a filtering problem since the 70s, but its evaluation for specific continuous-time, discrete-state systems remains a demanding task. As an advantage, Markov renewal processes (MrP) retain their renewal property under state space filtering. This offers a way to solve the filtering problem analytically for small systems. We consider a class of communication systems $X \to Y$ that can be derived from an MrP by a custom filtering procedure. For the subclasses, where (i) $Y$ is a renewal process or (ii) $(X,Y)$ belongs to a class of MrPs, we provide an evolution equation for finite transmission duration $T>0$ and limit theorems for $T \to \infty$ that facilitate simulation-free evaluation of the MI $\mathbb{I}(X_{[0,T]}; Y_{[0,T]})$ and its associated mutual information rate (MIR). In other cases, simulation cost is reduced to the marginal system $(X,Y)$ or $Y$. We show that systems with an additional $X$-modulating level $C$, which statically chooses between different processes $X_{[0,T]}(c)$, can naturally be included in our framework, thereby giving an expression for $\mathbb{I}(C; Y_{[0,T]})$. Our primary contribution is to apply the results of classical (Markov renewal) filtering theory in a novel manner to the problem of exactly computing the MI/MIR. The theoretical framework is showcased in an application to bacterial gene expression, where filtering is analytically tractable.