  In a Bayesian analysis, the likelihood that specific candidate parameters
govern the evolution of a quantum system are conditioned on the outcome of
measurements which, in turn, cause measurement backaction on the state of the
system [M. Tsang, Phys. Rev. Lett. 108, 170502 (2012)]. Specializing to the
distinction of two candidate hypotheses, we study the achievements of
continuous monitoring of the radiation emitted by a quantum system followed by
an optimal projective measurement on its conditioned final state. Our study of
the radiative decay of a driven two-level system shows an intricate interplay
between the maximum information available from photon counting and homodyne
detection and the final projective measurement on the emitter. We compare the
results with theory predicting a lower bound for the probability to assign a
wrong hypothesis by any combined measurement on the system and its radiative
environment.
