Introduction We present a scheme for learning probabilistic dependency grammars from positive training examples plus constraints on rules. In particular we present the results of two experiments. The first, in which the constraints were minimal, was unsuccessful. The second, with significant constraints, was successful within the bounds of the task we had set. We will explicate dependency grammars in Section 2. For the moment we simply note that they are a very restricted class of grammars which do not fit exactly into the Chomsky hierarchy, but whose appearance is most like the context-free grammars. We assume that the goal of learning a context-free grammar needs no justification. The problem has attracted a fair amount of attention, ( [1,4] are good surveys. ) but no good solutions have been found. Our choice of learning from only positive training examples needs only a little more justification. Obviously, if it is possible, a scheme which only uses positive training exampl