nLab Fisher metric

Fisher metric is a metric appearing in information geometry, see there for more information and references. There are also several quantum versions. One is the Bures metric and another is quantum Fisher information matrix.

• wikipedia quantum information, Bures metric

Most references about Fisher metric are at information geometry, here we add few on quantum and $C^*$-algebraic analogues.

• P. Facchi, R. Kulkarni, V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, F. Ventriglia, Classical and quantum Fisher information in the geometrical formulation of quantum mechanics, Physics Letters A 374 pp. 4801 (2010)doi

There is an analogue in free probability

• D.-V. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory. V: Noncommutative Hilbert transforms, Invent. Math. 132:1 (1998) 189–227.

For singular statistical models (including those arising in machine learning) one needs more version of Fisher metric beyond manifolds; one possibility is in the framework of diffeologies,

• Hông Vân Lê, Diffeological statistical models and diffeological Hausdorff measures, video yt, slides pdf
• Hông Vân Lê, Alexey A. Tuzhilin, Nonparametric estimations and the diffeological Fisher metric, arXiv:2011.13418
• Hông Vân Lê, Diffeological statistical models,the Fisher metric and probabilistic mappings, Mathematics 2020, 8(2),167, arXiv:1912.02090