Numerical methods and codes.-- et al. ; The XMM-Large scale structure (XMM-LSS), XMM-Cosmological evolution survey (XMM-COSMOS), and XMM-Chandra deep field south (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 point spread function. Therefore this is one of the best samples available to determine the 2-10 keV luminosity function of active galactic nuclei (AGN) and their 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 10-10 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, which is 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 luminosity and density evolution (LADE) or luminosity-dependent density evolution (LDDE), 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 recently proposed 15-parameter extended LDDE model and find that this extension is not supported by our data. The strength of our method is that it provides unabsorbed, non-parametric estimates, credible intervals for luminosity function parameters, and a model choice based on predictive power for future data. ; P.R. acknowledges a grant from the Greek General Secretariat of Research and Technology in the framework of the programme Support of Postdoctoral Researchers. F.J.C. acknowledges support by the Spanish ministry of Economy and Competitiveness through the grant AYA2012-31447. ; Peer Reviewed