total population comprising unobscured AGN and obscured Compton-thin AGN, but excluding Compton-thick obscured AGN) as a function of $L_{\rm{X}}$ (2–10 luminosity; ):

$$f_{\rm{2}} = [0.27 + \beta(log L_{\rm{X}} - 43.75)]g(z),$$

where $\beta = -0.281$ and $g(z)$ is evolution of the form (1+z)$^{0.62}$ out to $z=2$ and flat thereafter. The number of Compton-thick obscured AGN is not as well constrained, and in W08 we simply boosted the space density of the Ueda et al. (2003) luminosity function by 50 per cent in a notional allowance for them, but left the issue open for subsequent refinement in post-processing. Ueda et al. showed that a reasonable match to the X-ray background results if the number of Compton-thick AGN simply equals the number of Compton-thin obscured AGN. We thus assume that the abundance of Compton-thick AGN population is a factor $f_{\rm{Cthick}}$ times that of the combined population of unobscured and Compton-thin obscured AGN, where $f_{\rm{Cthick}}$ = min\[0.5,$f_{\rm{2}}$\], with the upper bound of 0.50 hard-wired into the W08 simulation. After the removal of any excess Compton-thick AGN, the remaining sources in the W08 catalogue are probabilistically identified as unobscured (type I), Compton-thin or Compton-thick obscured (type II) AGN. Finally, 10 per cent of all sources 