  A unifying formalism of generalized conditional expectations (GCEs) for
quantum mechanics has recently emerged, but its physical implications regarding
the retrodiction of a quantum observable remain controversial. To address the
controversy, here I offer operational meanings for a version of the GCEs in the
context of quantum parameter estimation. When a quantum sensor is corrupted by
decoherence, the GCE is found to relate the operator-valued optimal estimators
before and after the decoherence. Furthermore, the error increase, or regret,
caused by the decoherence is shown to be equal to a divergence between the two
estimators. The real weak value as a special case of the GCE plays the same
role in suboptimal estimation -- its divergence from the optimal estimator is
precisely the regret for not using the optimal measurement. For an application
of the GCE, I show that it enables the use of dynamic programming for designing
a controller that minimizes the estimation error. For the frequentist setting,
I show that the GCE leads to a quantum Rao-Blackwell theorem, which offers
significant implications for quantum metrology and thermal-light sensing in
particular. These results give the GCE and the associated divergence a natural,
useful, and incontrovertible role in quantum decision and control theory.
