Probabilistic DFA inference is the problem of inducing a stochastic regular grammar from a positive sample of an unknown language. The ALERGIA algorithm is one of the most successful approaches to this problem. In the present work we review this algorithm and explain why its generalization criterion, a state merging operation, is purely local. This characteristic leads to the conclusion that there is no explicit way to bound the divergence between the distribution de ned by the solution and the training set distribution (that is, to control globally the generalization from the training sample). In this paper we present an alternative approach, the MDI algorithm, in which the solution is a probabilistic automaton that trades o minimal divergence from the training sample and minimal size. An e cient computation of the Kullback-Leibler divergence between two probabilistic DFAs is described, from which the new learning criterion is derived. Empirical results in the d...