Aim: The goal of this work package is to derive an accurate model to describe the relation between the AMICO mass proxies, the cluster lensing shear profile and the underlying halo mass, which is effective over the whole mass range probed by the cosmological catalog. As anticipated this corresponds to characterize the probability distribution P(O|M,z), which encodes the scaling relation and survey selection function. Of paramount importance are: i) the modeling of the tails of the distribution, which controls the purity and completeness of the sample, and ii) the modeling of the correlations between mass proxies and weak lensing signal, which if not properly addressed leads to biases in the cluster mass estimates, and thus in the inferred cosmological parameters. At present, a fully empirical calibration of these relations is not feasible due to the lack of multi-wavelength data at low S/N, and thus requires a careful use of simulated data which accounts for the current level of systematic uncertainties. Moreover, the implementation of ever more sophisticated models to account for correlated observables and systematics effects made the computation of theoretical quantities highly time consuming (O(100s)) to use standard sampling techniques for parameter inference (e.g. [To22]). Along with the characterization of the model we aim at identifying and developing computational techniques which allow a fast and accurate derivation of the theoretical quantities required for the likelihood evaluation.