Informational and Causal Architecture of Discrete-Time Renewal Processes

Sarah Marzen

Redwood Center for Theoretical Neuroscience
Physics Department
University of California at Berkeley
Berkeley, CA 94720


James P. Crutchfield

Complexity Sciences Center
Mathematics Department
Physics Department
University of California at Davis
Davis, CA 95616

ABSTRACT: Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose 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 finite number of causal states despite having an unbounded interevent count distribution. We use these 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 ε-machine presentation. All in all, the results lay the groundwork for analyzing processes with infinite statistical complexity and infinite excess entropy.

Sarah Marzen and James P. Crutchfield, "Informational and Causal Architecture of Discrete-Time Renewal Processes", Entropy 17:7 (2014) 4891-4917. doi:10.3390/e17074891.
[pdf] 1.7 MB
Santa Fe Institute Working Paper 14-08-032. [cond-mat.stat-mech].