 <p>No. Generally, MRFs can represent arbitrary Gibbs distributions (see the <a href="https://en.wikipedia.org/wiki/Hammersley%E2%80%93Clifford_theorem" rel="nofollow">Hammersley-Clifford theorem</a>). This is broad class but doesn't encompass everything.</p> <p>The <em>pairwise</em> 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.</p> <p>Finally, even if they <em>could</em> 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.</p>