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Joan-Andreu Sánchez, José-Miguel Benedí. Consistency of Stocastic Context–Free Grammars from Probabilistic Estimation based on Growth transformation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 1997. Vol. 19 (9), pp. 1052-1055.

An important problem related to the probabilistic estimation of Stochastic Context-Free Grammars (SCFGs) is guaranteeing the consistency of the estimated model. This problem was considered in [Booth and Thompson, 1973 and Wetherell, 1980] and studied in [Maryanski, 1974 and Chaudhuri and Rao 1983] for unambiguous SCFGs only, when the probabilistic distributions were estimated by the relative frequencies in a training sample. In this work, we extend this result by proving that the property of consistency is guaranteed for all SCFGs without restrictions, when the probability distributions are learned from the classical Inside-Outside and Viterbi algorithms, both of which are based on Growth Transformations. Other important probabilistic properties which are related to these results are also proven.