Posterior probability distribution of the hyper-parameters of the extended model introduced in Sect. 5.1 given the mock data of a sample of 1000 lenses. Red lines show the fit to the whole dataset (image positions and radial magnification ratios). Filled contours show the fit to image position only. Contour levels correspond to 68% and 95% enclosed probability regions. Dashed lines indicate the true values of the hyper-parameters, which are defined by fitting the each model directly to the distribution of log M₂₀₀, γ_DM, and log α_sps of the mock sample.