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# On the Fisher metric of conditional probability polytopes

Publikationstyp

Journal Article

Publikationsdatum

2014-06-06

Sprache

English

Enthalten in

Volume

16

Issue

6

Start Page

3207

End Page

3233

Citation

Entropy 16 (6): 3207-3233 (2014)

Publisher DOI

Scopus ID

ArXiv ID

Publisher

MDPI

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.

Schlagworte

Conditional model

Convex support polytope

Fisher information metric

Information geometry

Isometric embedding

Markov morphism

Natural gradient

Mathematics - Differential Geometry

Mathematics - Differential Geometry

Mathematics - Statistics

Statistics - Theory

53C99

DDC Class

004: Informatik

510: Mathematik