1 Statistical NLP: Lecture 12 Probabilistic Context Free Grammars

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).

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)

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.

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.

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) ?

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

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.

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)

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)

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)

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)

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.

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.