https://pure.iiasa.ac.at/id/eprint/14521 2021-08-27T17:41:50Z info:eu-repo/semantics/descriptiveMetadata https://pure.iiasa.ac.at/id/eprint/14521/ Understanding frequency distributions of path-dependent processes with non-multinomial maximum entropy approaches Hanel, R. Corominas-Murtra, B. Thurner, S. 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. IOP 2017-03-06 Article PeerReviewed text en cc_by https://pure.iiasa.ac.at/id/eprint/14521/1/Hanel_2017_New_J._Phys._19_033008.pdf Hanel, R., Corominas-Murtra, B., & Thurner, S. <https://pure.iiasa.ac.at/view/iiasa/307.html> (2017). Understanding frequency distributions of path-dependent processes with non-multinomial maximum entropy approaches. New Journal of Physics 19 (3) e033008. 10.1088/1367-2630/aa611d <https://doi.org/10.1088/1367-2630%2Faa611d>. 10.1088/1367-2630/aa611d 10.1088/1367-2630/aa611d 10.1088/1367-2630/aa611d info:eu-repo/semantics/objectFile info:eu-repo/semantics/humanStartPage