Bayesian inversion and the Tomita–Takesaki modular group

Author:

Giorgetti Luca1234,Parzygnat Arthur J1234,Ranallo Alessio1234,Russo  Benjamin P1234

Affiliation:

1. Dipartimento di Matematica, Università di Roma Tor Vergata , Via della Ricerca Scientifica, 1, I-00133 Roma, Italy

2. Graduate School of Informatics, Nagoya University , Chikusa-Ku, Nagoya 464-8601, Japan

3. Dipartimento di Matematica, Università di Roma Tor Vergata , Via della Ricerca Scientifica, 1, I-00133 Roma, Italy

4. Mathematics in Computation Section, Computer Science and Mathematics Division, Oak Ridge National Laboratory , Oak Ridge, TN 37831, USA

Abstract

Abstract We show that conditional expectations, optimal hypotheses, disintegrations and adjoints of unital completely positive maps are all instances of Bayesian inverses. We study the existence of the latter by means of the Tomita–Takesaki modular group and we provide extensions of a theorem of Takesaki as well as a theorem of Accardi and Cecchini to the setting of not necessarily faithful states on finite-dimensional $C^{\ast}$-algebras.

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

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