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

TL;DR

This paper explores the information-geometric structure of finite causal-state processes and introduces a process-replicator dynamic that generalizes classical biological models within an evolutionary framework.

Contribution

It provides a mathematical framework linking information geometry with evolutionary dynamics of stochastic processes, including a novel generalization of the replicator equation.

Findings

Configuration space forms Riemannian manifolds with entropy-based metrics.

Evolutionary inference follows Riemannian gradient flow of fitness.

The dynamics generalize the Wright-Fisher and replicator models.

Abstract

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.