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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.15221 |
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| _version_ | 1866917694685577216 |
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| author | Gehri, Maximilian Engelmann, Nicolai Koeppl, Heinz |
| author_facet | Gehri, Maximilian Engelmann, Nicolai Koeppl, Heinz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mutual Information of a class of Poisson-type Channels using Markov Renewal Theory Gehri, Maximilian Engelmann, Nicolai Koeppl, Heinz Information 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. |
| title | Mutual Information of a class of Poisson-type Channels using Markov Renewal Theory |
| topic | Information Theory |
| url | https://arxiv.org/abs/2403.15221 |