Analytical computation of frequency distributions of path-dependent processes by means of a non-multinomial maximum entropy approach

Path-dependent stochastic processes are often non-ergodic and the ensemble picture can no longer be used to compute observables. The resulting mathematical difficulties impose limits to the analytical understanding of path-dependent processes. The statistics of path-dependent processes is often not multinomial, in the sense that the multiplicities for the occurrence of events is not a multinomial factor. The popular maximum entropy principle is tightly related to multinomial processes, non-interacting systems and to the ensemble picture, and loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, and can be constructed if the non-multinomial statistics of the underlying process is captured correctly in a functional that plays the role of an entropy. We demonstrate this for P\'olya urn processes that serve as a well known prototype for path-dependent processes. P\'olya urns, which have been used in a huge range of practical applications, are self-reinforcing processes that explicitly break multinomial structure. We show the predictive power of the resulting analytical method by computing the frequency and rank distributions of P\'olya urn processes. For the first time we are able to use microscopic update rules of a path-dependent process to construct a non-multinomial entropy functional, that, when maximized, yields details of time-dependent distribution functions, as we show numerically. We discuss the implications on the limits of max-ent based statistical inference when the statistics of a data source is unknown.