AGN are known to have complex X-ray spectra that depend on both the properties of the accreting SMBH (e.g. mass, accretion rate) and the distribution of obscuring material in its vicinity ("torus"). Often however, simple and even unphysical models are adopted to represent the X-ray spectra of AGN. In the case of blank field surveys in particular, this should have an impact on e.g. the determination of the AGN luminosity function, the inferred accretion history of the Universe and also on our understanding of the relation between AGN and their host galaxies. We develop a Bayesian framework for model comparison and parameter estimation of X-ray spectra. We take into account uncertainties associated with X-ray data and photometric redshifts. We also demonstrate how Bayesian model comparison can be used to select among ten different physically motivated X-ray spectral models the one that provides a better representation of the observations. Despite the use of low-count spectra, our methodology is able to draw strong inferences on the geometry of the torus. For a sample of 350 AGN in the 4 Ms Chandra Deep Field South field, our analysis identifies four components needed to represent the diversity of the observed X-ray spectra: (abridged). Simpler models are ruled out with decisive evidence in favour of a geometrically extended structure with significant Compton scattering. Regarding the geometry of the obscurer, there is strong evidence against both a completely closed or entirely open toroidal geometry, in favour of an intermediate case. The additional Compton reflection required by data over that predicted by toroidal geometry models, may be a sign of a density gradient in the torus or reflection off the accretion disk. Finally, we release a catalogue with estimated parameters such as the accretion luminosity in the 2-10 keV band and the column density, $N_{H}$, of the obscurer.

X-ray spectral modelling of the AGN obscuring region in the CDFS: Bayesian model selection and catalogue