<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities.</p><p>For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.</p></abstract>

The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities.For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.

Geometry of the probability simplex and its connection to the maximum entropy method

S.K. Yadav Sony Computer Science Laboratories

Journal of Applied Mathematics, Statistics and Informatics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian...

Geometry of the probability simplex and its connection to the maximum entropy method

Published Online: Jul 09, 2020

Page range: 25 - 35

DOI: https://doi.org/10.2478/jamsi-2020-0003

Keywords
Geometry on positive vectors, geometry on the probability simplex, logarithmic distance on the class of positive vectors, the maximum entropy method

© 2020 H. Gzyl et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Geometry of the probability simplex and its connection to the maximum entropy method

Published Online: Jul 09, 2020

Page range: 25 - 35

DOI: https://doi.org/10.2478/jamsi-2020-0003

KeywordsGeometry on positive vectors, geometry on the probability simplex, logarithmic distance on the class of positive vectors, the maximum entropy method

© 2020 H. Gzyl et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Geometry on positive vectors, geometry on the probability simplex, logarithmic distance on the class of positive vectors, the maximum entropy method