Generalized conformal structures on statistical manifolds

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K04842
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya Institute of Technology
• Principal Investigator: Matsuzoe Hiroshi 名古屋工業大学, 工学(系)研究科(研究院), 教授 (90315177)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 統計多様体 / 情報幾何学 / 共形射影構造 / エスコート分布

Outline of Final Research Achievements

A statistical manifold is a generalization of geometric structures naturally formulated for statistical models from the viewpoint of differential geometry, where a statistical model is a set of probability density functions that have suitable regularity conditions. On the other hand, a conformal structure is a geometrical structure for equivalence of metrics (or inner products) that preserve orthogonality and angle.
In this study, we show that statistical models such as the deformation exponential families admit sequential geometrical structures, and have a generalized conformal equivalence structures between invariant statistical manifolds and flat statistical manifolds.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

従来の情報幾何学は，指数型分布族に対して双対平坦空間の構造を議論するものが中心であった．本研究では近年重要性の増している非指数型の分布族に対して，その統計多様体の逐次構造を解明するとともに，不変統計多様体構造と平坦統計多様体構造の間の一般化した共形構造を解明した．統計多様体の幾何学の微分幾何学的基礎として非常に有意義な研究成果である．