Deformation of Fisher information and entropy in non-commutative probability spaces

2019 Fiscal Year Final Research Report

• Project/Area Number: 26400112
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Ochanomizu University
• Principal Investigator: Yoshida Hiroaki お茶の水女子大学, 基幹研究院, 教授 (10220667)
• Project Period (FY): 2014-04-01 – 2020-03-31
• Keywords: 非可換確率論 / 作用素環論 / 関数解析学 / エントロピー / 量子変形

Outline of Final Research Achievements

In this study, we have investigated quantum deformed entropy and Fisher information in non-commutative probability space. Using the correspondence between quantum deformed independence and potential functions, the free analogue of the beta prime distribution and its related distributions, F-distribution and t-distribution, have been introduced to the theory of free probability related to the score function of Fisher information.
Furthermore, we have succeeded to give an application of random matrices to the statistical data analysis, which allows us to expect further applications of non-commutative probability theory to other fields.

Free Research Field

非可換確率論

Academic Significance and Societal Importance of the Research Achievements

本研究における, 非可換確率空間における変形エントロピーならびに Fisher 情報量に関する調査により, 変形独立性が持つ多様性についての理解が深まり, 非可換確率論, 特にランダム行列を用いた自由確率論統計的データ解析などの他分野への応用が開拓された. 今後もさらに機械学習など, その応用分野の拡張が期待されるようになった.