@article{iiasa14521,
          volume = {19},
          number = {3},
           month = {March},
           title = {Understanding frequency distributions of path-dependent processes with non-multinomial maximum entropy approaches},
       publisher = {IOP},
            year = {2017},
         journal = {New Journal of Physics},
             doi = {10.1088/1367-2630/aa611d},
           pages = {e033008},
             url = {https://pure.iiasa.ac.at/id/eprint/14521/},
            issn = {1367-2630},
        abstract = {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 P{\'o}lya 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 P{\'o}lya 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.},
          author = {Hanel, R. and Corominas-Murtra, B. and Thurner, S.}
}