What cannot be learned with Bethe Approximations

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. However not much is known about the theoretical properties of such approximations. Here we show that there exists a regime of empirical marginals where such "Bethe learning" will fail. By failure we mean that moment matching will not be 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.