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On the Fisher metric of conditional probability polytopes
Publikationstyp
Journal Article
Date Issued
2014-06-06
Sprache
English
Author(s)
Journal
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.
Subjects
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