  The inference of a hidden variable's historical value, based on observations
before and after the fact, is a controversial subject in quantum mechanics.
Here I address the controversy by proposing a formalism that unifies and
generalizes some of the previous proposals for the task, including the quantum
minimum-mean-square-error estimators proposed by Ohki, the generalized
conditional expectation proposed by Accardi and Cecchini, the quantum smoothing
theory proposed by Tsang, the optimal observables for parameter estimation
proposed by Personick, Belavkin, and Grishanin, and the weak values proposed by
Aharonov, Albert, and Vaidman. The formalism is based on Ohki's suggestion of a
distance between two observables in the Heisenberg picture, which remains well
defined for incompatible observables and serves as a more general foundation
for quantum inference than Belavkin's nondemolition principle.
