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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.20035 |
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| _version_ | 1866912294530711552 |
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| author | Bian, Zheng Bollt, Erik M. |
| author_facet | Bian, Zheng Bollt, Erik M. |
| contents | We study conditional mutual information (cMI) between a pair of variables $X,Y$ given a third one $Z$ and derived quantities including transfer entropy (TE) and causation entropy (CE) in the dynamically relevant context where $X=T(Y,Z)$ is determined by $Y,Z$ via a deterministic transformation $T$. Under mild continuity assumptions on their distributions, we prove a zero-infinity dichotomy for cMI for a wide class of $T$, which gives a yes-or-no answer to the question of information flow as quantified by TE or CE. Such an answer fails to distinguish between the relative amounts of information flow. To resolve this problem, we propose a discretization strategy and a conjectured formula to discern the \textit{relative ambiguities} of the system, which can serve as a reliable proxy for the relative amounts of information flow. We illustrate and validate this approach with numerical evidence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The problem of infinite information flow Bian, Zheng Bollt, Erik M. Dynamical Systems Information Theory 94A17, 37C30 H.1.1 We study conditional mutual information (cMI) between a pair of variables $X,Y$ given a third one $Z$ and derived quantities including transfer entropy (TE) and causation entropy (CE) in the dynamically relevant context where $X=T(Y,Z)$ is determined by $Y,Z$ via a deterministic transformation $T$. Under mild continuity assumptions on their distributions, we prove a zero-infinity dichotomy for cMI for a wide class of $T$, which gives a yes-or-no answer to the question of information flow as quantified by TE or CE. Such an answer fails to distinguish between the relative amounts of information flow. To resolve this problem, we propose a discretization strategy and a conjectured formula to discern the \textit{relative ambiguities} of the system, which can serve as a reliable proxy for the relative amounts of information flow. We illustrate and validate this approach with numerical evidence. |
| title | The problem of infinite information flow |
| topic | Dynamical Systems Information Theory 94A17, 37C30 H.1.1 |
| url | https://arxiv.org/abs/2503.20035 |