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 (2010) 4801 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, Diffeological statistical models,the Fisher metric and probabilistic mappings, Mathematics 2020, 8(2) 167 arXiv:1912.02090

• Hông Vân Lê, Natural differentiable structures on statistical models and the Fisher metric, Information Geometry (2022) arXiv:2208.06539 doi

• Hông Vân Lê, Alexey A. Tuzhilin, Nonparametric estimations and the diffeological Fisher metric, In: Barbaresco F., Nielsen F. (eds) Geometric Structures of Statistical Physics, Information Geometry, and Learning, p. 120–138, SPIGL 2020. Springer Proceedings in Mathematics & Statistics 361, doi arXiv:2011.13418

In this paper, first, we survey the concept of diffeological Fisher metric and its naturality, using functorial language of probability morphisms, and slightly extending Lê’s theory in (Le2020) to include weakly $C^k$-diffeological statistical models. Then we introduce the resulting notions of the diffeological Fisher distance, the diffeological Hausdorff–Jeffrey measure and explain their role in classical and Bayesian nonparametric estimation problems in statistics.

category: probability