  We consider the system identification problem of estimating a dynamical
parameter of a Markovian quantum open system (the atom maser), by performing
continuous time measurements in the system's output (outgoing atoms). Two
estimation methods are investigated and compared. On the one hand, the maximum
likelihood estimator (MLE) takes into account the full measurement data and is
asymptotically optimal in terms of its mean square error. On the other hand,
the `likelihood-free' method of approximate Bayesian computation (ABC) produces
an approximation of the posterior distribution for a given set of summary
statistics, by sampling trajectories at different parameter values and
comparing them with the measurement data via chosen statistics.
  Our analysis is performed on the atom maser model, which exhibits interesting
features such as bistability and dynamical phase transitions, and has
connections with the classical theory of hidden Markov processes. Building on
previous results which showed that atom counts are poor statistics for certain
values of the Rabi angle, we apply MLE to the full measurement data and
estimate its Fisher information. We then select several correlation statistics
such as waiting times, distribution of successive identical detections, and use
them as input of the ABC algorithm. The resulting posterior distribution
follows closely the data likelihood, showing that the selected statistics
contain `most' statistical information about the Rabi angle.
