Reconstructing AGN X-ray spectral parameter distributions with Bayesian
  methods II: Population inference
  We present a new Bayesian method to reconstruct the parent distributions of
X-ray spectral parameters of active galactic nuclei (AGN) in large surveys,
using the probability distribution function (PDF) of posteriors obtained by
fitting a consistent physical model to each object with a Bayesian method,
presented in our companion paper. The PDFs are often broadly distributed and
some present systematic biases, hence naive point estimators or even some
standard parametric modelling are not sufficient to reconstruct the parent
population without obvious bias. Our method uses a redistribution matrix
computed from a large realistic simulation with the same selection as in the
actual sample to redistribute the stacked PDF, and then forward-fit a
non-parametric model to it in a Bayesian way, so that the biases in the PDFs
are properly taken into account. In this way, we are able to get accurate
uncertainties on the model parameters. We apply our spectral fitting and
population inference methods to the XMM-COSMOS survey as a pilot study. For the
819 AGN detected in the COSMOS field, 663 (81\%) of which having spectroscopic
redshifts (spec-z) and the others having high quality photometric redshifts
(photo-z), we find prominent bi-modality with widely separated peaks in the
distribution of the absorbing hydrogen column density ($N_\mathrm{H}$), and an
indication that absorbed AGN have harder photon indices. A clear decreasing
trend of the absorbed AGN fraction vs the intrinsic 2--10\,keV luminosity is
presented, but there is no clear evolution trend of the absorbed fraction with
redshift. Our method is designed to be readily applicable to large AGN samples
like XXL, and eventually eROSITA.
