Abstract: In this talk I will address the direct problem of the deep (restricted) Boltzmann machine (DBM) in the form of a multi-partite spin-glass model. Multi-partite models emerge as a natural generalization of mean field models where the permutation symmetry among the spins is weakened and preserved only inside each partition of the system. These models were studied both in deterministic and disordered settings and in general their solution requires a technical convexity assumption on the interactions that does not hold for the DBM. This feature makes it a challenging problem. Indeed, the DBM is still unsolved for centered interactions. On the contrary, in my talk I will show how to exactly and rigorously compute its thermodynamic limit on the Nishimori line, a particular subregion of the phase space where replica symmetry is proved to hold. As I will try to explain, besides its interest in Statistical Mechanics the Nishimori line also provides a bridge between disordered systems and Statistical Inference. Finally, I will discuss how the geometry of the model, especially the relative sizes of the layers, can influence the phase transition. This is expected to provide some insight on the optimal shape of a deep architecture in order to have a non trivial spontaneous magnetization.