Next Article in Journal
On Spatial Covariance, Second Law of Thermodynamics and Configurational Forces in Continua
Next Article in Special Issue
On Clustering Histograms with k-Means by Using Mixed α-Divergences
Previous Article in Journal
Relative Entropy, Interaction Energy and the Nature of Dissipation
Previous Article in Special Issue
Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions
Article Menu

Export Article

Open AccessArticle
Entropy 2014, 16(6), 3207-3233; doi:10.3390/e16063207

On the Fisher Metric of Conditional Probability Polytopes

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, Leipzig 04103, Germany
Department of Mathematics and Computer Science, Leipzig University, PF 10 09 20, Leipzig 04009, Germany
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Author to whom correspondence should be addressed.
Received: 31 March 2014 / Revised: 18 May 2014 / Accepted: 29 May 2014 / Published: 6 June 2014
(This article belongs to the Special Issue Information Geometry)
View Full-Text   |   Download PDF [354 KB, uploaded 24 February 2015]   |  


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. View Full-Text
Keywords: Fisher information metric; information geometry; convex support polytope; conditional model; Markov morphism; isometric embedding; natural gradient Fisher information metric; information geometry; convex support polytope; conditional model; Markov morphism; isometric embedding; natural gradient

This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

MDPI and ACS Style

Montúfar, G.; Rauh, J.; Ay, N. On the Fisher Metric of Conditional Probability Polytopes. Entropy 2014, 16, 3207-3233.

Show more citation formats Show less citations formats

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top