The mechanism driving the evolution and dispersal of protoplanetary discs has a substantial impact on the theories of planet formation. To date, neither the viscous paradigm nor the MHD wind-driven framework have been ruled out by the observations. The statistical richness of the observed samples provides a unique opportunity to test evolutionary models through population synthesis. However, comparisons between models and observations are normally carried out by eye, and a more statistically rigorous approach has not yet been implemented. We present DEMOS, a Bayesian inference tool that aims to infer the model parameters and initial conditions that best reproduce the full distribution of global properties of disc populations. This framework relies on a comparison between observations and synthetic populations generated with the 1D disc population-synthesis code Diskpop. We tested the tool with both the MHD wind model and the viscous framework with internal photoevaporation. Applying DEMOS to synthetic populations, we find that, for both models, it accurately recovers the model parameters of disc populations and their associated uncertainties, including cases in which initial conditions are correlated with the stellar mass. In the optimal scenario in which disc masses, accretion rates and disc sizes are available, the tool can also constrain the angular-momentum transport parameter. We show that, although parameter estimates are the most precise with samples of about 100 discs, the tool also yields meaningful constraints for smaller populations of 10-30 discs. We further find that the predictions obtained with DEMOS closely align with those found by previous works. We anticipate that DEMOS will be a valuable tool to infer the initial conditions of observed disc populations, to design observational programmes, and to assess which evolutionary model best reproduces the data.