  The XMM-LSS, XMM-COSMOS, and XMM-CDFS surveys are complementary in terms of
sky coverage and depth. Together, they form a clean sample with the least
possible variance in instrument effective areas and PSF. Therefore this is one
of the best samples available to determine the 2-10 keV luminosity function of
AGN and its evolution. The samples and the relevant corrections for
incompleteness are described. A total of 2887 AGN is used to build the LF in
the luminosity interval 10^42-10^46 erg/s, and in the redshift interval
0.001-4. A new method to correct for absorption by considering the probability
distribution for the column density conditioned on the hardness ratio is
presented. The binned luminosity function and its evolution is determined with
a variant of the Page-Carrera method, improved to include corrections for
absorption and to account for the full probability distribution of photometric
redshifts. Parametric models, namely a double power-law with LADE or LDDE
evolution, are explored using Bayesian inference. We introduce the
Watanabe-Akaike information criterion (WAIC) to compare the models and estimate
their predictive power. Our data are best described by the LADE model, as
hinted by the WAIC indicator. We also explore the 15-parameter extended LDDE
model recently proposed by Ueda et al., and find that this extension is not
supported by our data. The strength of our method is that it provides:
un-absorbed non-parametric estimates; credible intervals for luminosity
function parameters; model choice according to which one has more predictive
power for future data.
