Sequence Models. Hidden Markov Models (HMM) MaxEnt Markov Models (MEMM) Many slides from Michael Collins
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1 Sequence Models Hidden Markov Models (HMM) MaxEnt Markov Models (MEMM) Many slides from Michael Collins
2 Overview and HMMs I The Tagging Problem I Generative models, and the noisy-channel model, for supervised learning I Hidden Markov Model (HMM) taggers I I I Basic definitions Parameter estimation The Viterbi algorithm
3 Part-of-Speech Tagging INPUT: Profits soared at Boeing Co., easily topping forecasts on Wall Street, as their CEO Alan Mulally announced first quarter results. OUTPUT: Profits/N soared/v at/p Boeing/N Co./N,/, easily/adv topping/v forecasts/n on/p Wall/N Street/N,/, as/p their/poss CEO/N Alan/N Mulally/N announced/v first/adj quarter/n results/n./. N V P Adv Adj... = Noun =Verb = Preposition =Adverb =Adjective
4 Named Entity Recognition INPUT: Profits soared at Boeing Co., easily topping forecasts on Wall Street, as their CEO Alan Mulally announced first quarter results. OUTPUT: Profits soared at [Company Boeing Co.], easily topping forecasts on [Location Wall Street], as their CEO[Person Alan Mulally] announced first quarter results.
5 Named Entity Extraction as Tagging INPUT: Profits soared at Boeing Co., easily topping forecasts on Wall Street, as their CEO Alan Mulally announced first quarter results. OUTPUT: Profits/NA soared/na at/na Boeing/SC Co./CC,/NA easily/na topping/na forecasts/na on/na Wall/SL Street/CL,/NA as/na their/na CEO/NA Alan/SP Mulally/CP announced/na first/na quarter/na results/na./na NA SC CC SL CL... =Noentity =StartCompany =ContinueCompany =StartLocation =ContinueLocation
6 Our Goal Training set: 1 Pierre/NNP Vinken/NNP,/, 61/CD years/nns old/jj,/, will/md join/vb the/dt board/nn as/in a/dt nonexecutive/jj director/nn Nov./NNP 29/CD./. 2 Mr./NNP Vinken/NNP is/vbz chairman/nn of/in Elsevier/NNP N.V./NNP,/, the/dt Dutch/NNP publishing/vbg group/nn./. 3 Rudolph/NNP Agnew/NNP,/, 55/CD years/nns old/jj and/cc chairman/nn of/in Consolidated/NNP Gold/NNP Fields/NNP PLC/NNP,/, was/vbd named/vbn a/dt nonexecutive/jj director/nn of/in this/dt British/JJ industrial/jj conglomerate/nn./ ,219 It/PRP is/vbz also/rb pulling/vbg 20/CD people/nns out/in of/in Puerto/NNP Rico/NNP,/, who/wp were/vbd helping/vbg Huricane/NNP Hugo/NNP victims/nns,/, and/cc sending/vbg them/prp to/to San/NNP Francisco/NNP instead/rb./. I From the training set, induce a function/algorithm that maps new sentences to their tag sequences.
7 Two Types of Constraints Influential/JJ members/nns of/in the/dt House/NNP Ways/NNP and/cc Means/NNP Committee/NNP introduced/vbd legislation/nn that/wdt would/md restrict/vb how/wrb the/dt new/jj savings-and-loan/nn bailout/nn agency/nn can/md raise/vb capital/nn./. I Local : e.g., can is more likely to be a modal verb MD rather than a noun NN I Contextual : e.g., a noun is much more likely than a verb to follow a determiner I Sometimes these preferences are in conflict: The trash can is in the garage
8 Overview I The Tagging Problem I Generative models, and the noisy-channel model, for supervised learning I Hidden Markov Model (HMM) taggers I I I Basic definitions Parameter estimation The Viterbi algorithm
9 Supervised Learning Problems I We have training examples x (i),y (i) for i =1...m. Each x (i) is an input, each y (i) is a label. I Task is to learn a function f mapping inputs x to labels f(x)
10 Supervised Learning Problems I We have training examples x (i),y (i) for i =1...m. Each x (i) is an input, each y (i) is a label. I Task is to learn a function f mapping inputs x to labels f(x) I Conditional models: I I Learn a distribution p(y x) from training examples For any test input x, definef(x) = arg maxy p(y x)
11 Generative Models I We have training examples x (i),y (i) for i =1...m. Task is to learn a function f mapping inputs x to labels f(x).
12 Generative Models I We have training examples x (i),y (i) for i =1...m. Task is to learn a function f mapping inputs x to labels f(x). I Generative models: I I Learn a distribution p(x, y) from training examples Often we have p(x, y) =p(y)p(x y)
13 Generative Models I We have training examples x (i),y (i) for i =1...m. Task is to learn a function f mapping inputs x to labels f(x). I Generative models: I I Learn a distribution p(x, y) from training examples Often we have p(x, y) =p(y)p(x y) I Note: we then have where p(x) = P y p(y)p(x y) p(y x) = p(y)p(x y) p(x)
14 Decoding with Generative Models I We have training examples x (i),y (i) for i =1...m. Task is to learn a function f mapping inputs x to labels f(x). I Generative models: I I Learn a distribution p(x, y) from training examples Often we have p(x, y) =p(y)p(x y) I Output from the model: f(x) = argmax y = arg max y = arg max y p(y x) p(y)p(x y) p(x) p(y)p(x y)
15 Overview I The Tagging Problem I Generative models, and the noisy-channel model, for supervised learning I Hidden Markov Model (HMM) taggers I I I Basic definitions Parameter estimation The Viterbi algorithm
16 Hidden Markov Models I We have an input sentence x = x 1,x 2,...,x n (x i is the i th word in the sentence) I We have a tag sequence y = y 1,y 2,...,y n (y i is the i th tag in the sentence) I We ll use an HMM to define p(x 1,x 2,...,x n,y 1,y 2,...,y n ) for any sentence x 1...x n and tag sequence y 1...y n of the same length. I Then the most likely tag sequence for x is arg max y 1...y n p(x 1...x n,y 1,y 2,...,y n )
17 Trigram Hidden Markov Models (Trigram HMMs) For any sentence x 1...x n where x i 2 V for i =1...n, and any tag sequence y 1...y n+1 where y i 2 S for i =1...n, and y n+1 = STOP, the joint probability of the sentence and tag sequence is p(x 1...x n,y 1...y n+1 )= n+1 Y q(y i y i 2,y i 1 ) ny e(x i y i ) i=1 i=1 where we have assumed that xy_0 0 = y_-1 x = 1 *. = *. Parameters of the model: I q(s u, v) for any s 2 S [ {STOP}, u, v 2 S [ {*} I e(x s) for any s 2 S, x 2 V
18 An Example If we have n =3, x 1...x 3 equal to the sentence the dog laughs, and y 1...y 4 equal to the tag sequence D N V STOP, then p(x 1...x n,y 1...y n+1 ) = q(d, ) q(n, D) q(v D, N) q(stop N, V) e(the D) e(dog N) e(laughs V) I STOP is a special tag that terminates the sequence I We take y 0 = y 1 = *, where * is a special padding symbol
19 Why the Name? p(x 1...x n,y 1...y n ) = q(stop y n 1,y n ) ny q(y j y j 2,y j 1 ) j=1 {z } Markov Chain ny e(x j y j ) j=1 {z } x j s are observed
20 Overview I The Tagging Problem I Generative models, and the noisy-channel model, for supervised learning I Hidden Markov Model (HMM) taggers I I I Basic definitions Parameter estimation The Viterbi algorithm
21 Smoothed Estimation q(vt DT, JJ) = Count(Dt, JJ, Vt) 1 Count(Dt, JJ) Count(JJ, Vt) + 2 Count(JJ) + 3 Count(Vt) Count() =1, and for all i, i 0 e(base Vt) = Count(Vt, base) Count(Vt)
22 Dealing with Low-Frequency Words: An Example Profits soared at Boeing Co., easily topping forecasts on Wall Street, as their CEO Alan Mulally announced first quarter results.
23 Dealing with Low-Frequency Words A common method is as follows: I Step 1: Split vocabulary into two sets Frequent words = wordsoccurring 5timesintraining Low frequency words =allotherwords I Step 2: Map low frequency words into a small, finite set, depending on prefixes, su xes etc.
24 Dealing with Low-Frequency Words: An Example [Bikel et. al 1999] (named-entity recognition) Word class Example Intuition twodigitnum 90 Two digit year fourdigitnum 1990 Four digit year containsdigitandalpha A Product code containsdigitanddash Date containsdigitandslash 11/9/89 Date containsdigitandcomma 23, Monetary amount containsdigitandperiod 1.00 Monetary amount, percentage othernum Other number allcaps BBN Organization capperiod M. Person name initial firstword first word of sentence no useful capitalization information initcap Sally Capitalized word lowercase can Uncapitalized word other, Punctuation marks, all other words
25 Dealing with Low-Frequency Words: An Example Profits/NA soared/na at/na Boeing/SC Co./CC,/NA easily/na topping/na forecasts/na on/na Wall/SL Street/CL,/NA as/na their/na CEO/NA Alan/SP Mulally/CP announced/na first/na quarter/na results/na./na firstword/na soared/na at/na initcap/sc Co./CC,/NA easily/na lowercase/na forecasts/na on/na initcap/sl Street/CL,/NA as/na their/na CEO/NA Alan/SP initcap/cp announced/na first/na quarter/na results/na./na NA =Noentity SC =StartCompany CC =ContinueCompany SL =StartLocation CL =ContinueLocation... +
26 Inference and the Viterbi Algorithm
27 The Viterbi Algorithm Problem: for an input x 1...x n, find arg max y 1...y n+1 p(x 1...x n,y 1...y n+1 ) where the arg max is taken over all sequences y 1...y n+1 such that y i 2 S for i =1...n, and y n+1 = STOP. We assume that p again takes the form p(x 1...x n,y 1...y n+1 )= n+1 Y q(y i y i 2,y i 1 ) ny e(x i y i ) i=1 i=1 Recall that we have assumed in this definition that y 0 = y 1 = *, and y n+1 = STOP.
28 Brute Force Search is Hopelessly Ine cient Problem: for an input x 1...x n, find arg max y 1...y n+1 p(x 1...x n,y 1...y n+1 ) where the arg max is taken over all sequences y 1...y n+1 such that y i 2 S for i =1...n, and y n+1 = STOP.
29 The Viterbi Algorithm I Define n to be the length of the sentence I Define S k for k = 1...n to be the set of possible tags at position k: S 1 = S 0 = { } I Define S k = S r(y 1,y 0,y 1,...,y k )= for k 2 {1...n} ky q(y i y i 2,y i 1 ) i=1 I Define a dynamic programming table ky e(x i y i ) i=1 (k, u, v) = maximum probability of a tag sequence ending in tags u, v at position k that is, (k, u, v) = max hy 1,y 0,y 1,...,y k i:y k 1 =u,y k =v r(y 1,y 0,y 1...y k )
30 A Recursive Definition Base case: (0, *, *) =1 Recursive definition: For any k 2 {1...n}, for any u 2 S k 1 and v 2 S k : (k, u, v) = max w2s k 2 ( (k 1,w,u) q(v w, u) e(x k v))
31 The Viterbi Algorithm Input: asentencex 1...x n,parametersq(s u, v) and e(x s). Initialization: Set (0, *, *) =1 Definition: S 1 = S 0 = { }, S k = S for k 2 {1...n} Algorithm: I For k =1...n, I For u 2 Sk 1, v 2 S k, (k, u, v) = max ( (k 1,w,u) q(v w, u) e(x k v)) w2s k 2 I Return max u2sn 1,v2S n ( (n, u, v) q(stop u, v))
32 The Viterbi Algorithm with Backpointers Input: asentencex 1...x n,parametersq(s u, v) and e(x s). Initialization: Set (0, *, *) =1 Definition: S 1 = S 0 = { }, S k = S for k 2 {1...n} Algorithm: I For k =1...n, I For u 2 Sk 1, v 2 S k, (k, u, v) = max w2s k 2 ( (k 1,w,u) q(v w, u) e(x k v)) bp(k, u, v) = arg max w2s k 2 ( (k 1,w,u) q(v w, u) e(x k v)) I Set (y n 1,y n ) = arg max (u,v) ( (n, u, v) q(stop u, v)) I For k =(n 2)...1, y k = bp(k +2,y k+1,y k+2 ) I Return the tag sequence y 1...y n
33 The Viterbi Algorithm: Running Time I O(n S 3 ) time to calculate q(s u, v) e(x k s) for all k, s, u, v. I n S 2 entries in to be filled in. I O( S ) time to fill in one entry I ) O(n S 3 ) time in total
34 A Simple Bi-gram Example: (X, Y): P(X/Y), POS tags for bears fish? noun *.80 bears noun.02 Verb *.10 bears verb.02 STOP noun.50 fish verb.07 STOP verb.50 fish noun.08 noun verb.77 verb noun.65 noun noun.0001 nerb verb.0001
35 Answer bears: noun fish: verb
36 Forward The Viterbi Algorithm Input: asentencex 1...x n,parametersq(s u, v) and e(x s). Initialization: Set (0, *, *) =1 Definition: S 1 = S 0 = { }, S k = S for k 2 {1...n} Algorithm: I For k =1...n, I For u 2 Sk 1, v 2 S k, (k, u, v) = Sum max ( (k 1,w,u) q(v w, u) e(x k v)) w2s k 2 Sum I Return max u2sn 1,v2S n ( (n, u, v) q(stop u, v))
37 Pros and Cons I Hidden markov model taggers are very simple to train (just need to compile counts from the training corpus) If you already have a labeled training set. Use forward-backward algorithms in the unsupervised setting. I Perform relatively well (over 90% performance on named entity recognition) I Main di culty is modeling e(word tag) can be very di cult if words are complex
38 MaxEnt Markov Models (MEMMs)
39 Log-Linear Models for Tagging I We have an input sentence w [1:n] = w 1,w 2,...,w n (w i is the i th word in the sentence) I We have a tag sequence t [1:n] = t 1,t 2,...,t n (t i is the i th tag in the sentence) I We ll use an log-linear model to define p(t 1,t 2,...,t n w 1,w 2,...,w n ) for any sentence w [1:n] and tag sequence t [1:n] of the same length. (Note: contrast with HMM that defines p(t 1...t n,w 1...w n )) I Then the most likely tag sequence for w [1:n] is t [1:n] = argmax t [1:n] p(t [1:n] w [1:n] )
40 How to model p(t [1:n] w [1:n] )? A Trigram Log-Linear Tagger: p(t [1:n] w [1:n] )= Q n j=1 p(t j w 1...w n,t 1...t j 1 ) Chain rule = Q n j=1 p(t j w 1,...,w n,t j 2,t j 1 ) Independence assumptions I We take t 0 = t 1 = * I Independence assumption: each tag only depends on previous two tags p(t j w 1,...,w n,t 1,...,t j 1 )=p(t j w 1,...,w n,t j 2,t j 1 )
41 An Example Hispaniola/NNP quickly/rb became/vb an/dt important/jj base/?? from which Spain expanded its empire into the rest of the Western Hemisphere. There are many possible tags in the position?? Y = {NN, NNS, Vt, Vi, IN, DT,... }
42 Representation: Histories I A history is a 4-tuple ht 2,t 1,w [1:n],ii I t 2,t 1 are the previous two tags. I w [1:n] are the n words in the input sentence. I i is the index of the word being tagged I X is the set of all possible histories Hispaniola/NNP quickly/rb became/vb an/dt important/jj base/?? from which Spain expanded its empire into the rest of the Western Hemisphere. I t 2,t 1 = DT, JJ I w [1:n] = hhispaniola, quickly, became,..., Hemisphere,.i I i =6
43 Recap: Feature Vector Representations in Log-Linear Models I We have some input domain X, and a finite label set Y. Aim is to provide a conditional probability p(y x) for any x 2 X and y 2 Y. I A feature is a function f : X Y! R (Often binary features or indicator functions f : X Y! {0, 1}). I Say we have m features f k for k =1...m ) A feature vector f(x, y) 2 R m for any x 2 X and y 2 Y.
44 f 1 (hjj, DT, h Hispaniola,... i, 6i, Vt) =1 f 2 (hjj, DT, h Hispaniola,... i, 6i, Vt) =0... An Example (continued) I X is the set of all possible histories of form ht 2,t 1,w [1:n],ii I Y = {NN, NNS, Vt, Vi, IN, DT,... } I We have m features f k : X Y! R for k =1...m For example: f 1 (h, t) = f 2 (h, t) =... 1 if current word wi is base and t = Vt 0 otherwise 1 if current word wi ends in ing and t = VBG 0 otherwise
45 Training the Log-Linear Model I To train a log-linear model, we need a training set (x i,y i ) for i =1...n. Then search for 0 1 v =argmax v X B log p(y i x i ; i {z } Log Likelihood (see last lecture on log-linear models) X vk 2 2 C k A {z } Regularizer I Training set is simply all history/tag pairs seen in the training data
46 The Viterbi Algorithm Problem: for an input w 1...w n, find arg max t 1...t n p(t 1...t n w 1...w n ) We assume that p takes the form p(t 1...t n w 1...w n )= ny i=1 q(t i t i 2,t i 1,w [1:n],i) (In our case q(t i t i 2,t i 1,w [1:n],i) is the estimate from a log-linear model.)
47 The Viterbi Algorithm I Define n to be the length of the sentence I Define r(t 1...t k )= ky i=1 q(t i t i 2,t i 1,w [1:n],i) I Define a dynamic programming table (k, u, v) = maximum probability of a tag sequence ending in tags u, v at position k that is, (k, u, v) = max r(t 1...t k 2, u, v) ht 1,...,t k 2 i
48 A Recursive Definition Base case: (0, *, *) =1 Recursive definition: For any k 2 {1...n}, for any u 2 S k 1 and v 2 S k : (k, u, v) = max t2s k 2 (k 1,t,u) q(v t, u, w [1:n],k) where S k is the set of possible tags at position k
49 The Viterbi Algorithm with Backpointers Input: asentencew 1...w n, log-linear model that provides q(v t, u, w [1:n],i) for any tag-trigram t, u, v, foranyi 2 {1...n} Initialization: Set (0, *, *) =1. Algorithm: I For k =1...n, I For u 2 Sk 1, v 2 S k, (k, u, v) = max t2s k 2 (k 1,t,u) q(v t, u, w [1:n],k) bp(k, u, v) = arg max t2s k 2 (k 1,t,u) q(v t, u, w [1:n],k) I Set (t n 1,t n ) = arg max (u,v) (n, u, v) I For k =(n 2)...1, t k = bp(k +2,t k+1,t k+2 ) I Return the tag sequence t 1...t n
50 Summary I Key ideas in log-linear taggers: I Decompose p(t 1...t n w 1...w n )= ny p(t i t i 2,t i 1,w 1...w n ) i=1 I Estimate p(t i t i 2,t i 1,w 1...w n ) I using a log-linear model For a test sentence w1...w n, use the Viterbi algorithm to find! ny arg max p(t i t i 2,t i 1,w 1...w n ) t 1...t n i=1 I Key advantage over HMM taggers: flexibility in the features they can use
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