TY - JOUR

T1 - Statistical Einstein manifolds of exponential families with group-invariant potential functions

AU - Peng, L.

AU - Zhang, Zhenning

N1 - Funding Information:
LP is partially supported by JSPS Grant-in-Aid for Scientific Research (No. 16KT0024 ), the MEXT ‘Top Global University Project’, Waseda University Grant for Special Research Projects (No. 2019C-179 , No. 2019E-036 , No. 2019R-081 ) and Waseda University Grant Program for Promotion of International Joint Research. ZZ is supported by Science and Technology Program of Beijing Municipal Commission of Education (No. KM201810005006 ).
Funding Information:
LP is partially supported by JSPS Grant-in-Aid for Scientific Research (No. 16KT0024), the MEXT ‘Top Global University Project’, Waseda University Grant for Special Research Projects (No. 2019C-179, No. 2019E-036, No. 2019R-081) and Waseda University Grant Program for Promotion of International Joint Research. ZZ is supported by Science and Technology Program of Beijing Municipal Commission of Education (No. KM201810005006).
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2019/11/15

Y1 - 2019/11/15

N2 - This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, information geometry was developed to understand the intrinsic properties of statistical models, which play important roles in statistical inference, etc. Among all these models, exponential families is one of the most important kinds, whose geometric structures are fully determined by their potential functions. To classify statistical Einstein manifolds, we derive partial differential equations for potential functions of exponential families; special solutions of these equations are obtained through the ansatz method as well as group-invariant solutions via reductions using Lie point symmetries.

AB - This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. A statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. With the Fisher information metric as a Riemannian metric, information geometry was developed to understand the intrinsic properties of statistical models, which play important roles in statistical inference, etc. Among all these models, exponential families is one of the most important kinds, whose geometric structures are fully determined by their potential functions. To classify statistical Einstein manifolds, we derive partial differential equations for potential functions of exponential families; special solutions of these equations are obtained through the ansatz method as well as group-invariant solutions via reductions using Lie point symmetries.

KW - Einstein manifold

KW - Group-invariant solutions

KW - Information geometry

KW - Symmetry reduction

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U2 - 10.1016/j.jmaa.2019.07.043

DO - 10.1016/j.jmaa.2019.07.043

M3 - Article

AN - SCOPUS:85069199179

SN - 0022-247X

VL - 479

SP - 2104

EP - 2118

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -