  We aim to constrain the evolution of AGN as a function of obscuration using
an X-ray selected sample of $\sim2000$ AGN from a multi-tiered survey including
the CDFS, AEGIS-XD, COSMOS and XMM-XXL fields. The spectra of individual X-ray
sources are analysed using a Bayesian methodology with a physically realistic
model to infer the posterior distribution of the hydrogen column density and
intrinsic X-ray luminosity. We develop a novel non-parametric method which
allows us to robustly infer the distribution of the AGN population in X-ray
luminosity, redshift and obscuring column density, relying only on minimal
smoothness assumptions. Our analysis properly incorporates uncertainties from
low count spectra, photometric redshift measurements, association
incompleteness and the limited sample size. We find that obscured AGN with
$N_{H}>{\rm 10^{22}\, cm^{-2}}$ account for ${77}^{+4}_{-5}\%$ of the number
density and luminosity density of the accretion SMBH population with $L_{{\rm
X}}>10^{43}\text{ erg/s}$, averaged over cosmic time. Compton-thick AGN account
for approximately half the number and luminosity density of the obscured
population, and ${38}^{+8}_{-7}\%$ of the total. We also find evidence that the
evolution is obscuration-dependent, with the strongest evolution around
$N_{H}\thickapprox10^{23}\text{ cm}^{-2}$. We highlight this by measuring the
obscured fraction in Compton-thin AGN, which increases towards $z\sim3$, where
it is $25\%$ higher than the local value. In contrast the fraction of
Compton-thick AGN is consistent with being constant at $\approx35\%$,
independent of redshift and accretion luminosity. We discuss our findings in
the context of existing models and conclude that the observed evolution is to
first order a side-effect of anti-hierarchical growth.
