Markov maps in noncommutative probability theory and mathematical statistics. (English) Zbl 0654.46058

Morozova, E. A.; Chentsov, N. N.

Probability theory and mathematical statistics, Proc. 4th Conf., Vilnius/USSR 1985, Vol. 2, 287-310 (1987).

[For the entire collection see Zbl 0637.00008.]
Following the approach of the monograph of the second author [Statistical decision rules and optimal inference (1982; Zbl 0484.62008)] certain aspects of “noncommutative” mathematical statistics are treated in terms of categorical geometry. In the basic category objects are full operator algebras (on Hilbert spaces) with a family of normal states and morphisms are completely positive normal unital mappings. Monotone invariants, loss functions and Riemann metrics are studied in this context.

Reviewer: D.Petz

MSC:

46L51	Noncommutative measure and integration
46L53	Noncommutative probability and statistics
46L54	Free probability and free operator algebras
62C05	General considerations in statistical decision theory

Keywords:

completely positive; Markov map; invariant metric; Markov category; noncommutative mathematical statistics; categorical geometry