  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.
