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PreprintCC BY 4.0arXiv

On the Information Bottleneck for Ergodic Finite-State Markov Chains: Stationary Approximation and Asymptotic Optimality

  • Wael Tabbara
  • Jihad Fahs
  • Mohamad Assaad
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Abstract

In this paper, we investigate the information bottleneck problem for sources that follow discrete-time ergodic finite-state Markov chains. We propose treating observations generated by the time-varying Markov chain as if they were drawn from its unique stationary distribution, and we use the solution to the information bottleneck problem for the stationary process to compress samples from the time-varying process. The resulting error is upper bounded by several information gap terms. We establish sufficient conditions under which these gaps converge to zero and analyze the convergence rates of the different information gap terms. The proofs are based on intricate analysis using convergence of subsequences along with the derivation of properties of the optimization sets associated with the information bottleneck problem. In addition to the inequalities and convergence results we establish, our analysis offers insights into how simple properties of ergodic Markov chains give rise to results concerning the information bottleneck problem for such chains.

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Cite

BibTeXRIS
BibTeX
@misc{tabbara2026information,
  title = {{On the Information Bottleneck for Ergodic Finite-State Markov Chains: Stationary Approximation and Asymptotic Optimality}},
  author = {Wael Tabbara and Jihad Fahs and Mohamad Assaad},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25952},
  eprint = {2610.25952},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25952},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Information Bottleneck for Ergodic Finite-State Markov Chains: Stationary Approximation and Asymptotic Optimality
AU  - Wael Tabbara
AU  - Jihad Fahs
AU  - Mohamad Assaad
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.25952
UR  - https://arxiv.org/abs/2610.25952
AB  - In this paper, we investigate the information bottleneck problem for sources that follow discrete-time ergodic finite-state Markov chains. We propose treating observations generated by the time-varying Markov chain as if they were drawn from its unique stationary distribution, and we use the solution to the information bottleneck problem for the stationary process to compress samples from the time-varying process. The resulting error is upper bounded by several information gap terms. We establish sufficient conditions under which these gaps converge to zero and analyze the convergence rates of the different information gap terms. The proofs are based on intricate analysis using convergence of subsequences along with the derivation of properties of the optimization sets associated with the information bottleneck problem. In addition to the inequalities and convergence results we establish, our analysis offers insights into how simple properties of ergodic Markov chains give rise to results concerning the information bottleneck problem for such chains.
ER  -