The longest-established important application of statistical techniques to linguistic problems is stylometry, a method of resolving disputed authorship (usually in a literary context, occasionally for forensic purposes) by finding statistical properties of text that are characteristic of individual writers, such as mean word or sentence length, or frequencies of particular words (see e.g. Morton 1978).
Such a technique was proposed by Augustus De Morgan in 1851 in connexion with the authorship of the Pauline Epistles; concrete investigations, using methods of greater or lesser statistical sophistication, have been carried out by many scholars, beginning with T.C. Mendenhall in 1887. The mathematical foundations of the topic were studied by Gustav Herdan (e.g. Herdan 1966), who showed for instance that the right way to measure vocabulary richness (type/token ratios), avoiding dependence on size of sample, is to divide log(types) by log(tokens). A.Q. Morton's stylometric demonstration that no more than five Epistles can be attributed to Paul attracted considerable attention in 1963, from a public intrigued by the idea that a computer (seen at that time as an obscure scientific instrument) might yield findings of religious significance. However, this work remains controversial. Other leading investigations were carried out by George Udny Yule on De Imitatione Christi, by Alvar Ellegård on the Letters of Junius, and by Frederick Mosteller and David Wallace on the Federalist Papers.
More recently, Douglas Biber (e.g. Biber 1995) has moved stylometry away from concern with individual authorship puzzles to shed light on broader considerations of historical evolution of style, and on genre differences in English and other languages, by means of factor analysis applied to grammatical features of texts.
A very different area is the use of numerical measures of resemblance between pairs of languages in order to establish the shape of the Stammbaum (family tree) of relationships among languages of a family. Early work (e.g. by Ernst Förstemann in 1852 and Jan Czekanowski in 1927) used phonological and grammatical properties to measure resemblance; but writers such as Morris Swadesh in the 1950s considered mainly the proportions of words for "core concepts" which remained cognate in pairs of descendant languages. The terms lexicostatistics and glottochronology are both used to describe such research (on which see e.g. Embleton 1986), although "glottochronology" is often reserved for work which assumes a constant rate of vocabulary replacement (an assumption regarded by many as unwarranted), and which thereby yields absolute dates for language-separation events. This type of research has recently been placed on a sounder theoretical footing by drawing on the axioms of biological cladistics. Don Ringe and others have used cladistic techniques to investigate relationships between the main branches of the Indo-European language family; in many respects their work confirms traditional views, but it gives unexpected results for the Germanic branch (which includes English). Johanna Nichols (1992) has used a different biologically-inspired approach, applying cluster-analysis techniques adapted from population genetics to statistics concerning the geographical incidence of grammatical features, in order to investigate early, large-scale human migration patterns. She claims that language statistics demonstrate that the Americas must have been colonized much earlier than was previously believed.
A third statistical approach to language was based on a generalization about word frequencies, stated by J.B. Estoup as early as 1916, but later publicized by George Kingsley Zipf, with whose name it is nowadays usually associated. Zipf's Law, , says that, in a long text, the rank of a word (commonest word = 1, second commonest = 2, etc.), multiplied by the word's frequency, will give roughly the same figure for all words. (Zipf also stated a second law, which asserted a log-linear relationship between word frequencies, and numbers of words sharing a particular frequency; he believed this to be a corollary of his first law, but that is incorrect - mathematically the two generalizations are independent of one another.) Zipf explained his "law" in terms of a Principle of Least Effort, on which he based a wide-ranging (and in some respects controversial) theory of human behaviour.
Although called a "law", Zipf's finding is an approximation only; commonly it is quite inaccurate for the highest and lowest ranks, though reasonably correct in the middle ranges. In the 1950s, Benoît Mandelbrot modified Zipf's Law to make it more empirically adequate, setting this work in the context of a proposed new discipline of "macrolinguistics", which was intended to bear the same relation to grammar (or "microlinguistics") as thermodynamics bears to the mechanics of individual gas molecules (Apostel et al. 1957). It is debatable whether Zipf's Law tells us anything surprising or deep about the nature of language or linguistic behaviour; George Miller argued in 1957 that the "law" was virtually a statistical necessity. But, as Christopher Manning and Hinrich Schütze have put it, Zipf's Law does remain a good way to summarize the insight that "what makes frequency-based approaches to language hard is that almost all words are rare".
In the 1960s and 1970s, numerical techniques fell out of favour in linguistics. (William Labov's pioneering work on the statistical rules governing social variation in language was carried out in conscious opposition to prevailing orthodoxies.) The subsequent revival of numerical approaches was due partly to a swing of intellectual fashion, but important roles were also played by two concrete developments: the creation of large samples or "corpora" of language in computer-readable form, the first of which was published as early as 1964, and the greatly increased accessibility of computers from the early 1980s. These factors made it easy for linguists to examine statistical properties of language which it would previously have been impractical to study.
Some corpus-based statistical work is an extension of traditional descriptive linguistics. Here are three examples (among many which could have been quoted). Our understanding of lexical differences between British and American English has been enlarged by Hofland & Johansson's publication of comprehensive tables of statistically significant differences between the frequencies of words in the two dialects (Hofland & Johansson 1982): these show, for instance, that masculine words such as he, boy, man, words referring to military or violent concepts, but also words from the religious domain, are significantly more frequent in American English, while feminine words, hedging words such as but and possible, words referring to family relationships, and the word disarmament, are significantly commoner in British English. (At least, this was true forty years ago, when the corpora analysed by Hofland & Johansson were compiled.) Harald Baayen has demonstrated that objective statistical methods for measuring the relative productivity of various word-derivation processes in English give results which sometimes contradict both linguists' intuitions and non-statistical linguistic theory. Sampson (2001, ch. 5) finds that the grammatical complexity of speech seems to increase with age of the speaker, not just in childhood but through middle and old age, contrary to what might be predicted from "critical period" theories of language learning.
Alongside investigations with purely scholarly goals, corpus-based statistical techniques have also come into play as an alternative to rule-based techniques in language engineering. (For a survey, see Manning & Schütze (1999).) A central function in many automatic language processing systems is parsing, i.e. grammatical analysis. The classic approach to automatic parsing of natural language treated the task as essentially similar to that of "compiling" computer programs written in a formal language, with a parser being derived from a generative grammar defining the class of legal inputs. However, this approach encounters difficulties if natural languages fail in practice to obey rigorous grammatical rules, or if (as is often the case) the rules allow very large numbers of alternative parses for sentences of only moderate length. Accordingly, the 1990s saw an upsurge of interest in parsing techniques based on language models which include statistical information distilled from corpora. For instance, probabilities may be assigned in various ways to the alternative rewrites allowed by a phrase-structure grammar; or the language model may altogether eschew the concept "ill-formed structure" (and "ill-formed string") and assign numerical scores to all possible ways of drawing a labelled tree over any word-string, with the correct parse scoring higher than alternatives in the case of those strings which are good sentences. Various (deterministic or stochastic) optimizing techniques are then used to locate the best-scoring analysis available for an input; and statistical optimizing techniques may also be used to induce the probabilistic grammar or language model automatically from data, which may take the form of a "treebank" - a language sample in which sentences have been equipped with their correct parse trees.
These approaches have brought computational linguistics into a closer relationship with speech research, where it has been recognized since the 1970s that the messiness and unpredictability of speech signals make probabilistic optimizing techniques centrally important for automatic speech recognition. In speech recognition (see e.g. Jelinek 1997), one is given a physical speech signal and aims to locate the sequence of words which is most probable, given that signal. In other words, one wishes to maximize a conditional probability p(w|s); and empirical data give one estimates for the converse conditional probabilities p(s|w), that is the probabilities of particular signals, given various word-sequences. Various useful language-engineering applications can be cast in similar terms. By Bayes's Theorem, ; so, in order to calculate the quantity we are interested in, we need to be able to estimate the p(w)'s, the prior probabilities of various word sequences in the language. Speech researchers commonly estimate p(w)'s using crude "n-gram" (usually bigram or trigram) language models, which estimate the probability of a long word-sequence simply from the frequencies of the overlapping pairs or triples of words comprised in the sequence, ignoring its linguistic structure. Linguists' hopes that grammar could be deployed to improve on the performance of n-gram models have not so far borne fruit. Conversely, simple collocation-based techniques have begun to be used for tasks that were previously assumed to depend crucially on subtle linguistic analysis: for instance, there have been experiments in automatic development of machine translation systems by extracting parallel collocations from large bodies of bilingual text, such as the Canadian Hansard. In general, one consequence of the introduction of statistical techniques in natural language processing has been a shift towards simpler grammar formalisms or language models than those which were popular in linguistics previously.
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