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.
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.