Fields of Application of Information Geometry

Abstract

1. Complexity measures can be geometrically built by using the information distance (Kullback–Leibler divergence) from families with restricted statistical dependencies. The Pythagorean geometry developed in Chaps. 2 and 4 allows us to iterate such constructions, by going to simpler and simpler families and taking—possibly weighted—sums, thereby systematically recovering many complexity measures proposed in the literature. 2. Evolutionary dynamics, as given by replicator equations or the Wright–Fisher model of population genetics, can also be naturally interpreted in the framework of information geometry. The fact that the coefficients of the Kolmogorov equations for the continuum limit of the Wright–Fisher model are given by the inverse Fisher metric, also known as the Shashahani metric in mathematical biology, allows us to systematically derive a wide range of formulas of population genetics from a unified geometric perspective. 3. Monte Carlo schemes can be interpreted as gradient flows w.r.t. the Fisher metric, and the relation between Langevin and Hamiltonian Monte Carlo can be understood in geometric terms. 4. As an outlook, infinite dimensional Gibbs families, as occurring for instance in quantum field theory, are put into the framework of information geometry in a heuristic manner.