On the probabilistic description of a multipartite correlation scenario with arbitrary numbers of settings and outcomes per site

We consistently formalize the probabilistic description of multipartite joint measurements performed on systems of any nature. This allows us (1) to specify in probabilistic terms the difference between nonsignaling, the Einstein-Podolsky-Rosen (EPR) locality and Bell's locality; (2) to introduce the notion of a local hidden variable (LHV) model for an S(1) x ... x S(N)-setting N-partite correlation experiment with outcomes of any spectral type, discrete or continuous, and to prove both general and specifically 'quantum' statements on an LHV simulation in an arbitrary multipartite case; (3) to classify LHV models for amultipartite quantum state, in particular, to show that any N-partite quantum state, pure or mixed, admits an arbitrary S(1) x 1 x ... x S(N)-setting LHV description; (4) to evaluate a threshold visibility for an arbitrary bipartite noisy quantum state to admit an S(1) x S(2)-setting LHV description under any generalized quantum measurements of two parties.