### Abstract

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.

Original language | English |
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Pages (from-to) | 2104-2118 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 479 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2019 Nov 15 |

### Fingerprint

### Keywords

- Einstein manifold
- Group-invariant solutions
- Information geometry
- Symmetry reduction

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Statistical Einstein manifolds of exponential families with group-invariant potential functions.** / Peng, Linyu; Zhang, Zhenning.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 479, no. 2, pp. 2104-2118. https://doi.org/10.1016/j.jmaa.2019.07.043

}

TY - JOUR

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

AU - Peng, Linyu

AU - Zhang, Zhenning

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

UR - http://www.scopus.com/inward/record.url?scp=85069199179&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069199179&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2019.07.043

DO - 10.1016/j.jmaa.2019.07.043

M3 - Article

AN - SCOPUS:85069199179

VL - 479

SP - 2104

EP - 2118

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -