  We address the problem of learning the parameters in graphical models when
inference is intractable. A common strategy in this case is to replace the
partition function with its Bethe approximation. We show that there exists a
regime of empirical marginals where such Bethe learning will fail. By failure
we mean that the empirical marginals cannot be recovered from the approximated
maximum likelihood parameters (i.e., moment matching is not achieved). We
provide several conditions on empirical marginals that yield outer and inner
bounds on the set of Bethe learnable marginals. An interesting implication of
our results is that there exists a large class of marginals that cannot be
obtained as stable fixed points of belief propagation. Taken together our
results provide a novel approach to analyzing learning with Bethe
approximations and highlight when it can be expected to work or fail.
