  Compared with the quantum trajectory equation, the quantum Bayesian approach
has the advantage of being more efficient to infer quantum state under
monitoring, based on the integrated output of measurement. For weak measurement
of qubits in circuit quantum electrodynamics(cQED), properly accounting for the
measurement backaction effects within the Bayesian framework is an important
problem of current interest.Elegant work towards this task was carried out by
Korotkov in "bad-cavity" and weak-response limits (arXiv:1111.4016).In the
present work, based on insights from the cavity-field states (dynamics) and the
help of an effective QTE, we generalize the results of arXiv:1111.4016 to more
general system parameters.The obtained Bayesian rule is in full agreement with
Korotkov's result in limiting cases and as well holds satisfactory accuracy in
non-limiting cases in comparison with the QTE simulations. We expect the
proposed Bayesian rule to be useful for future cQED measurement and control
experiments.
