  We study two free energy approximations (Bethe and plaquette-CVM) for the
Random Field Ising Model in two dimensions. We compare results obtained by
these two methods in single instances of the model on the square grid, showing
the difficulties arising in defining a robust critical line. We also attempt
average case calculations using a replica-symmetric ansatz, and compare the
results with single instances. Both, Bethe and plaquette-CVM approximations
present a similar panorama in the phase space, predicting long range order at
low temperatures and fields. We show that plaquette-CVM is more precise, in the
sense that predicts a lower critical line (the truth being no line at all).
Furthermore, we give some insight on the non-trivial structure of the fixed
points of different message passing algorithms.
