On the Fisher Metric of Conditional Probability Polytopes

Entropy

Guido Mont﫺r¹  Johannes Rauh¹ 

[1] Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, Leipzig 04103, Germany; E-Mails:

关键词: Fisher information metric;    information geometry;    convex support polytope;    conditional model;    Markov morphism;    isometric embedding;    natural gradient;   

DOI  :  10.3390/e16063207

来源: mdpi

【 摘 要 】

We consider three different approaches to define natural Riemannian metrics on polytopes of stochastic matrices. First, we define a natural class of stochastic maps between these polytopes and give a metric characterization of Chentsov type in terms of invariance with respect to these maps. Second, we consider the Fisher metric defined on arbitrary polytopes through their embeddings as exponential families in the probability simplex. We show that these metrics can also be characterized by an invariance principle with respect to morphisms of exponential families. Third, we consider the Fisher metric resulting from embedding the polytope of stochastic matrices in a simplex of joint distributions by specifying a marginal distribution. All three approaches result in slight variations of products of Fisher metrics. This is consistent with the nature of polytopes of stochastic matrices, which are Cartesian products of probability simplices. The first approach yields a scaled product of Fisher metrics; the second, a product of Fisher metrics; and the third, a product of Fisher metrics scaled by the marginal distribution.

【 授权许可】

CC BY   
© 2014 by the authors; licensee MDPI, Basel, Switzerland