Abstract
In this paper we investigate statistical manifolds with almost quaternionic structures. We define the concept of quaternionic Kähler-like statistical manifold and derive the main properties of quaternionic Kähler-like statistical submersions, extending in a new setting some previous results obtained by K. Takano concerning statistical manifolds endowed with almost complex and almost contact structures. Finally, we give a nontrivial example and propose some open problems in the field for further research.
Highlights
It is well known that the concept of statistical manifold arises naturally from divergencies—likeKullback–Leibler relative entropy—in statistics, information theory and related fields [1,2]
We investigate very natural kind of statistical manifold, namely those endowed with almost quaternionic structures, extending the results of K
It is well known there is a deep relationship between statistics and differential geometry
Summary
Kullback–Leibler relative entropy—in statistics, information theory and related fields [1,2]. In [10] the author considers the statistical model of the multivariate normal distribution as the Riemannian manifold and constructs an interesting example of statistical submersion. We remark that a complex version of the notion of statistical structure was considered in [11], where the author derived a condition for the curvature of a statistical manifold to admit a kind of standard hypersurface. The existence of symplectic structures on statistical manifolds was investigated in [12], where the author obtained a duality relation between the Fubini–Study metric on a projective space and the Fisher metric on a statistical model on a finite set. We investigate very natural kind of statistical manifold, namely those endowed with almost quaternionic structures, extending the results of K. This paper ends with conclusions and several open problems in the field for further research
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