An n-partite graph has the problem that edges can still only connect two vertices at a time. I think you would also need to include hyper-edges (faces, solids, etc). The joint probabilities would be represented by these hyper-edges, and the bounding ("lower") edges represent progressively more marginal probabilities. Higher category theory studies these kinds of objects, so it wouldn't particularly surprise me.

I wonder if simplicial complexes would work here? https://en.wikipedia.org/wiki/Simplicial_complex