  This report presents some fundamental mathematical results towards
elucidating the information-geometric underpinnings of evolutionary modelling
schemes for (quasi-)stationary discrete stochastic processes. The model class
under consideration is that of finite causal-state processes, known from the
computational mechanics programme, along with their minimal unifilar hidden
Markov generators. The respective configuration space is exhibited as a
collection of combinatorially related Riemannian manifolds wherein the metric
tensor field is an infinitesimal version of the relative entropy rate.
Furthermore, a certain evolutionary inference iteration is defined which can be
executed by generator-carrying agents and generalizes the Wright-Fisher model
from population genetics. The induced dynamics on configuration space is
studied from the large deviation point of view and it is shown that the
associated asymptotic expectation dynamics follows the Riemannian gradient flow
of a given fitness potential. In fact, this flow can formally be viewed as an
information-geometric generalization of the replicator dynamics from population
biology.
