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

On the Fisher Metric of Conditional Probability Polytopes

1,* , 1
 and 1,2,3
1 Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, Leipzig 04103, Germany 2 Department of Mathematics and Computer Science, Leipzig University, PF 10 09 20, Leipzig 04009, Germany 3 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)
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Abstract

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.
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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

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