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This report presents some fundamental mathematical results towardselucidating the information-geometric underpinnings of evolutionary modellingschemes for (quasi-)stationary discrete stochastic processes. The model classunder consideration is that of finite causal-state processes, known from thecomputational mechanics programme, along with their minimal unifilar hiddenMarkov generators. The respective configuration space is exhibited as acollection of combinatorially related Riemannian manifolds wherein the metrictensor field is an infinitesimal version of the relative entropy rate.Furthermore, a certain evolutionary inference iteration is defined which can beexecuted by generator-carrying agents and generalizes the Wright-Fisher modelfrom population genetics. The induced dynamics on configuration space isstudied from the large deviation point of view and it is shown that theassociated asymptotic expectation dynamics follows the Riemannian gradient flowof a given fitness potential. In fact, this flow can formally be viewed as aninformation-geometric generalization of the replicator dynamics from populationbiology.

Finitary Process Evolution I: Information Geometry of Configuration Space and the Process-Replicator Dynamics