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MiS Preprint
87/2014

On the Fisher Metric of Conditional Probability Polytopes

Guido Montúfar, Johannes Rauh and Nihat Ay

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.

Received:
19.08.2014
Published:
20.08.2014
MSC Codes:
53C99
Keywords:
Fisher information metric, information geometry, convex support polytope, conditional model, Markov morphism, isometric embedding, natural gradient

Related publications

inJournal
2014 Journal Open Access
Guido Montúfar, Johannes Rauh and Nihat Ay

On the Fisher metric of conditional probability polytopes

In: Entropy, 16 (2014) 6, pp. 3207-3233