Bayesian inversion and the Tomita–Takesaki modular group
- Resource Type
- Authors
- Giorgetti, Luca; Parzygnat, Arthur J.; Ranallo, Alessio; Russo, Benjamin P.
- Source
- The Quarterly Journal of Mathematics.
- Subject
- Quantum Physics
Mathematics::Operator Algebras
General Mathematics
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
Mathematics - Operator Algebras
FOS: Physical sciences
[PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph]
Bayesian
47A05, 47D03, 81P47
[PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph]
Settore MAT/05
FOS: Mathematics
group
modular
Operator Algebras (math.OA)
Quantum Physics (quant-ph)
[PHYS.QPHY] Physics [physics]/Quantum Physics [quant-ph]
- Language
- ISSN
- 1464-3847
0033-5606
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.