1-Conformal geometry of quasi statistical manifolds
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1-Conformal geometry of quasi statistical manifolds Keisuke Haba1 Received: 13 July 2019 / Revised: 20 September 2020 / Accepted: 25 September 2020 © Springer Nature Singapore Pte Ltd. 2020
Abstract A quasi statistical manifold is a generalization of a statistical manifold. The notion of quasi statistical manifolds was introduced to formulate the geometry of nonconservative estimating functions in statistics. Later, it was showed that quasi statistical manifolds are induced from affine distributions in the same way as statistical manifolds are induced from affine immersions. Here, an affine distribution is a non-integrable version of an affine immersion, and it is useful in quantum information geometry. On the other hand, it is known that generalized conformal geometry is useful for the study of statistical manifolds from the viewpoint of affine differential geometry. In particular, 1-conformal geometry of statistical manifolds gives a relation with the notion of affine immersions. Although generalized conformal geometry of quasi statistical manifolds is also expected to be useful, the geometry has not been cleared yet. The aim of this paper is to formulate 1-conformal geometry of quasi statistical manifolds. We research a relation between 1-conformal geometry of quasi statistical manifolds and the notion of affine distributions. As the main result, we show the fundamental theorems for affine distributions. We also formulate a hypersurface theory of quasi statistical manifolds. Keywords Information geometry · Statistical manifold admitting torsion · Quasi statistical manifold · Affine distribution · 1-Conformal geometry
1 Introduction A statistical manifold is an important geometric structure in information geometry, and it appears various fields in mathematical sciences [1]. A statistical manifold was originally introduced by Lauritzen [9]. He defined this structure by the triplet (M, g, D), where M is a smooth manifold, g is a Riemannian metric, and D is a symmetric (0, 3)-tensor field on M. Later, Kurose reformulated the definition of statistical mani-
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Keisuke Haba [email protected] Toyota Systems Corporation, Toyota, Japan
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Information Geometry
folds from the viewpoint of affine differential geometry [6]. He defined this structure by the triplet (M, ∇, h), where ∇ is a torsion-free affine connection on M, and h is a (pseudo)-Riemannian metric such that a (0, 3)-tensor field ∇h is symmetric. This definition is essentially equivalent to Lauritzen’s one [11]. In this paper, we follow Kurose’s definition. Kurose and Matsuzoe have studied generalized conformal geometry of statis tical manifolds [6,7,10]. Two statistical manifolds (M, ∇, h) and (M, ∇, h) are conformally-projectively equivalent if there exist smooth functions φ and ψ on M such that h(X , Y ) = eφ+ψ h(X , Y ), X Y = ∇ X Y + dφ(Y )X + dφ(X )Y − h(X , Y )gradh ψ, ∇ where gradh ψ is the gradient vector field of ψ defined by h(X , gradh ψ) = dψ(X ). The statistical manifold (M, ∇, h) is conformally-projectively flat if it is
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