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Informational and Causal Architecture of Discrete-Time Renewal Processes

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Journal Article - Open Access

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Department of Physics, University of California at Berkeley Berkeley United States

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Renewal processes are broadly used to model stochastic behavior consistingof isolated events separated by periods of quiescence, whose durations are specified by agiven probability law. Here, we identify the minimal sufficient statistic for their predictionthe set of causal states, calculate the historical memory capacity required to store thosestates statistical complexity, delineate what information is predictable excess entropy, anddecompose the entropy of a single measurement into that shared with the past, future, or both.The causal state equivalence relation defines a new subclass of renewal processes with a finitenumber of causal states despite having an unbounded interevent count distribution. We usethe resulting formulae to analyze the output of the parametrized Simple Nonunifilar Source,generated by a simple two-state hidden Markov model, but with an infinite-state -machinepresentation. All in all, the results lay the groundwork for analyzing more complex processeswith infinite statistical complexity and infinite excess entropy.

Subject Categories:

  • Information Science
  • Theoretical Mathematics
  • Statistics and Probability
  • Operations Research

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