The Effect of Non-tightness on Bayesian Estimation of PCFGs

Probabilistic context-free grammars have the unusual property of not always defining tight distributions (i.e., the sum of the ``probabilities'' of the trees the grammar generates can be less than one). This paper reviews how this non-tightness can arise and discusses its impact on Bayesian estimation of PCFGs. We begin by presenting the notion of ``almost everywhere tight grammars'' and show that linear CFGs follow it. We then propose three different ways of reinterpreting non-tight PCFGs to make them tight, show that the Bayesian estimators in Johnson et al. (2007) are correct under one of them, and provide MCMC samplers for the other two. We conclude with a discussion of the impact of tightness empirically.