  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\'olya urn
processes, which explicitly generalise multinomial statistics. We demonstrate
the adequacy of this constructive approach towards non-multinomial pendants of
entropy by computing frequency and rank distributions of P\'olya 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.
