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Path-dependent stochastic processes are often non-ergodic and observables canno longer be computed within the ensemble picture. The resulting mathematicaldifficulties pose severe limits to the analytical understanding ofpath-dependent processes. Their statistics is typically non-multinomial in thesense that the multiplicities of the occurrence of states is not a multinomialfactor. The maximum entropy principle is tightly related to multinomialprocesses, non-interacting systems, and to the ensemble picture; It loses itsmeaning for path-dependent processes. Here we show that an equivalent to theensemble picture exists for path-dependent processes, such that thenon-multinomial statistics of the underlying dynamical process, byconstruction, is captured correctly in a functional that plays the role of arelative entropy. We demonstrate this for self-reinforcing P\'olya urnprocesses, which explicitly generalise multinomial statistics. We demonstratethe adequacy of this constructive approach towards non-multinomial pendants ofentropy by computing frequency and rank distributions of P\'olya urn processes.We show how microscopic update rules of a path-dependent process allow us toexplicitly construct a non-multinomial entropy functional, that, whenmaximized, predicts the time-dependent distribution function.

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