Estimation of Probabilistic Context-Free Grammars

The assignment of probabilities to the productions of a context-free grammar may generate an improper distribution: the probability of all finite parse trees is less than one. The condition for proper assignment is rather subtle. Production probabilities can be estimated from parsed or unparsed sentences, and the question arises as to whether or not an estimated system is automatically proper. We show here that estimated production probabilities always yield proper distributions.