Hanel R, Corominas-Murtra B, & Thurner S (2017). Understanding frequency distributions of path-dependent processes with non-multinomial maximum entropy approaches. New Journal of Physics 19 (3): e033008. DOI:10.1088/1367-2630/aa611d.
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Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. Their statistics is typically non-multinomial in the sense that the multiplicities of the occurrence of states is not a multinomial factor. The maximum entropy principle is tightly related to multinomial processes, non-interacting systems, and to the ensemble picture; it loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, such that the non-multinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of a relative entropy. We demonstrate this for self-reinforcing Pólya urn processes, which explicitly generalize multinomial statistics. We demonstrate the adequacy of this constructive approach towards non-multinomial entropies by computing frequency and rank distributions of Pólya urn processes. We show how microscopic update rules of a path-dependent process allow us to explicitly construct a non-multinomial entropy functional, that, when maximized, predicts the time-dependent distribution function.
|Research Programs:||Advanced Systems Analysis (ASA)|
|Depositing User:||Luke Kirwan|
|Date Deposited:||05 Apr 2017 07:04|
|Last Modified:||05 Apr 2017 12:54|
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