Measuring Grammatical Diversity from Small Corpora: Derivational Entropy Rates, Mean Length of Utterances, and Annotation Invariance

In many fields, such as language acquisition, neuropsychology of language, the study of aging, and historical linguistics, corpora are used for estimating the diversity of grammatical structures that are produced during a period by an individual, community, or type of speakers. In these cases, treebanks are taken as representative samples of the syntactic structures that might be encountered. Generalizing the potential syntactic diversity from the structures documented in a small corpus requires careful extrapolation whose accuracy is constrained by the limited size of representative sub-corpora. In this article, I demonstrate -- theoretically, and empirically -- that a grammar's derivational entropy and the mean length of the utterances (MLU) it generates are fundamentally linked, giving rise to a new measure, the derivational entropy rate. The mean length of utterances becomes the most practical index of syntactic complexity; I demonstrate that MLU is not a mere proxy, but a fundamental measure of syntactic diversity. In combination with the new derivational entropy rate measure, it provides a theory-free assessment of grammatical complexity. The derivational entropy rate indexes the rate at which different grammatical annotation frameworks determine the grammatical complexity of treebanks. I introduce the Smoothed Induced Treebank Entropy (SITE) as a tool for estimating these measures accurately, even from very small treebanks. I conclude by discussing important implications of these results for both NLP and human language processing.