Deepening and development of information geometric methods for statistical science

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K11872
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 60030:Statistical science-related
• Research Institution: The Institute of Statistical Mathematics
• Principal Investigator: Henmi Masayuki 統計数理研究所, 統計基盤数理研究系, 准教授 (80465921)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 無限次元情報幾何

Outline of Final Research Achievements

The result of this research is as follows. Most of the studies on infinite-dimensional manifolds of probability density functions have been conducted from a mathematical point of view. First of all, we considered the two existing researches based on Orlicz spaces and Hilbert spaces and showed that they are insufficient as a framework for geometry of semiparametric inference in statistics. In addition, we proposed a road map to introduce statistically meaningful structure of an infinite-dimensional manifold based on the concepts of differentiability of a parameter functional, tangent spaces and so on, which are defined without mentioning manifold structure in the theory of semiparametric inference.

Free Research Field

統計科学

Academic Significance and Societal Importance of the Research Achievements

情報幾何学では、確率密度関数の集合としての統計モデルが有限次元多様体として扱える場合についてはよく研究されていて、統計学にも応用されているが、セミ(ノン)パラメトリックな統計手法などを幾何学的に議論するためには、統計モデルが「無限次元」となる場合の理論を整備する必要がある。本研究での成果はその出発点に過ぎないが、その理論の整備は、情報幾何学の応用範囲を大きく広げ、統計学の発展にも寄与するものである。