Can a fully connected, pairwise graphical model estimate an arbitrary joint distribution on N binary variables?
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No. Generally, MRFs can represent arbitrary Gibbs distributions (see the Hammersley-Clifford theorem). This is broad class but doesn't encompass everything.
The pairwise constraint is further limiting. So far as I can tell, not all MRFs with higher-order potentials can be represented by a pairwise MRF, so it stands to reason that a pairwise MRF cannot represent an arbitrary distribution.
Finally, even if they could represent an arbitrary joint distribution, it would be a moot point for MRFs of any reasonable size - exact inference is going to be massively intractable, so you'd be constrained to whatever assumptions your approximation would make.