Bayesian inversion and the Tomita-Takesaki modular group

TL;DR

This paper explores the connection between Bayesian inverses and modular theory in operator algebras, extending foundational theorems to broader contexts involving non-faithful states on finite-dimensional C*-algebras.

Contribution

It demonstrates that various concepts like conditional expectations and adjoints are instances of Bayesian inverses and extends key theorems using the Tomita-Takesaki modular group.

Findings

Unified framework for Bayesian inverses in operator algebras

Extended Takesaki and Accardi-Cecchini theorems

Applicability to non-faithful states on finite-dimensional C*-algebras

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 $C^*$-algebras.