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1 Statistical NLP: Lecture 12 Probabilistic Context Free Grammars

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2 Motivation 4 N-gram models and HMM Tagging only allowed us to process sentences linearly. 4 However, even simple sentences require a nonlinear model that reflects the hierarchical structure of sentences rather than the linear order of words. 4 Probabilistic Context Free Grammars are the simplest and most natural probabilistic model for tree structures and the algorithms for them are closely related to those for HMMs. 4 Note, however, that there are other ways of building probabilistic models of syntactic structure (see Chapter 12).

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3 Formal Definition of PCFGs A PCFG consists of: –A set of terminals, {w k }, k= 1,…,V –A set of nonterminals, N i, i= 1,…, n –A designated start symbol N 1 –A set of rules, {N i --> j }, (where j is a sequence of terminals and nonterminals) –A corresponding set of probabilities on rules such that: i j P(N i --> j ) = 1 4 The probability of a sentence (according to grammar G) is given by:. P(w 1m, t) where t is a parse tree of the sentence. = {t: yield(t)=w1m} P(t)

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4 Assumptions of the Model 4 Place Invariance: The probability of a subtree does not depend on where in the string the words it dominates are. 4 Context Free: The probability of a subtree does not depend on words not dominated by the subtree. 4 Ancestor Free: The probability of a subtree does not depend on nodes in the derivation outside the subtree.

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5 Some Features of PCFGs 4 A PCFG gives some idea of the plausibility of different parses. However, the probabilities are based on structural factors and not lexical ones. 4 PCFG are good for grammar induction. 4 PCFGs are robust. 4 PCFGs give a probabilistic language model for English. 4 The predictive power of a PCFG tends to be greater than for an HMM. Though in practice, it is worse. 4 PCFGs are not good models alone but they can be combined with a tri-gram model. 4 PCFGs have certain biases which may not be appropriate.

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6 Questions fo PCFGs 4 Just as for HMMs, there are three basic questions we wish to answer: 4 What is the probability of a sentence w 1m according to a grammar G: P(w 1m |G)? 4 What is the most likely parse for a sentence: argmax t P(t|w 1m,G)? 4 How can we choose rule probabilities for the grammar G that maximize the probability of a sentence, argmax G P(w1m|G) ?

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7 Restriction 4 In this lecture, we only consider the case of Chomsky Normal Form Grammars, which only have unary and binary rules of the form: N i --> N j N k N i --> w j 4 The parameters of a PCFG in Chomsky Normal Form are: P(N j --> N r N s | G), an n3 matrix of parameters P(N j --> w k |G), nV parameters (where n is the number of nonterminals and V is the number of terminals) 4 r,s P(N j --> N r N s ) + k P (N j --> w k ) =1

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8 From HMMs to Probabilistic Regular Grammars (PRG) 4 A PRG has start state N 1 and rules of the form: –N i --> w j N k –N i --> w j This is similar to what we had for an HMM except that in an HMM, we have n w1n P(w 1n ) = 1 whereas in a PCFG, we have w L P(w) = 1 where L is the language generated by the grammar. 4 PRG are related to HMMs in that a PRG is a HMM to which we should add a start state and a finish (or sink) state.

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9 From PRGs to PCFGs 4 In the HMM, we were able to efficiently do calculations in terms of forward and backward probabilities. 4 In a parse tree, the forward probability corresponds to everything above and including a certain node, while the backward probability corresponds to the probability of everything below a certain node. 4 We introduce Outside ( j ) and Inside ( j ) Probs.: – j (p,q)=P(w 1(p-1), N pq j,w (q+1)m |G) – j (p,q)=P(w pq |N pq j, G)

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10 The Probability of a String I: Using Inside Probabilities 4 We use the Inside Algorithm, a dynamic programming algorithm based on the inside probabilities: P(w 1m |G) = P(N 1 ==>* w 1m |G) =. P(w 1m |N 1m 1, G)= 1 (1,m) 4 Base Case: j (k,k) = P(w k |N kk j, G)=P(N j --> w k |G) 4 Induction: j (p,q) = r,s d=p q-1 P(N j --> N r N s ) r (p,d) s (d+1,q)

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11 The Probability of a String II: Using Outside Probabilities 4 We use the Outside Algorithm based on the outside probabilities: P(w 1m |G)= j j (k,k)P(N j --> w k ) 4 Base Case: 1 (1,m)= 1; j (1,m)=0 for j 1 4 Inductive Case: j (p,q)=. 4 Similarly to the HMM, we can combine the inside and the outside probabilities: P(w 1m, N pq |G)= j j (p,q) j (p,q)

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12 Finding the Most Likely Parse for a Sentence 4 The algorithm works by finding the highest probability partial parse tree spanning a certain substring that is rooted with a certain nonterminal. 4 i (p,q) = the highest inside probability parse of a subtree N pq i 4 Initialization: i (p,p) = P(Ni --> w p ) 4 Induction: i (p,q) = max 1 j,k n,p r N j N k ) j (p,r) k (r+1,q) 4 Store backtrace: i (p,q)=argmax (j,k,r) P(N i --> N j N k ) j (p,r) k (r+1,q) 4 Termination: P(t ^ )= 1 (1,m)

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13 Training a PCFG 4 Restrictions: We assume that the set of rules is given in advance and we try to find the optimal probabilities to assign to different grammar rules. 4 Like for the HMMs, we use an EM Training Algorithm called the Inside-Outside Algorithm which allows us to train the parameters of a PCFG on unannotated sentences of the language. 4 Basic Assumption: a good grammar is one that makes the sentences in the training corpus likely to occur ==> we seek the grammar that maximizes the likelihood of the training data.

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14 Problems with the Inside-Outside Algorithm 4 Extremely Slow: For each sentence, each iteration of training is O(m 3 n 3 ). 4 Local Maxima are much more of a problem than in HMMs 4 Satisfactory learning requires many more nonterminals than are theoretically needed to describe the language. 4 There is no guarantee that the learned nonterminals will be linguistically motivated.

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