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 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
$T$
that facilitate simulation-free evaluation of the MI 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 (c), can naturally be included in our framework, thereby giving an expression for 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.