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 an LHV model for an S_{1}x...xS_{N}-setting N-partite correlation experiment, with outcomes of any spectral type, discrete or continuous, and to prove both general and specific "quantum" statements on an LHV simulation in an arbitrary multipartite case; (3) to classify LHV models for a multipartite quantum state, in particular, to show that any N-partite quantum state, pure or mixed, admits an Sx1x...x1 -setting LHV description; (4) to evaluate a threshold visibility for a noisy bipartite quantum state to admit an S_{1}xS_ {2}-setting LHV description under any generalized quantum measurements of two parties. In a sequel to this paper, we shall introduce a single general representation incorporating in a unique manner all Bell-type inequalities for either joint probabilities or correlation functions that have been introduced or will be introduced in the literature.