Language Modeling. Michael Collins, Columbia University
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1 Language Modeling Michael Collins, Columbia University
2 Overview The language modeling problem Trigram models Evaluating language models: perplexity Estimation techniques: Linear interpolation Discounting methods
3 The Language Modeling Problem We have some (finite) vocabulary, say V = {the, a, man, telescope, Beckham, two,...} We have an (infinite) set of strings, V the STOP a STOP the fan STOP the fan saw Beckham STOP the fan saw saw STOP the fan saw Beckham play for Real Madrid STOP
4 The Language Modeling Problem (Continued) We have a training sample of example sentences in English
5 The Language Modeling Problem (Continued) We have a training sample of example sentences in English We need to learn a probability distribution p i.e., p is a function that satisfies x V p(x) = 1, p(x) 0 for all x V
6 The Language Modeling Problem (Continued) We have a training sample of example sentences in English We need to learn a probability distribution p i.e., p is a function that satisfies x V p(x) = 1, p(x) 0 for all x V p(the STOP) = p(the fan STOP) = 10 8 p(the fan saw Beckham STOP) = p(the fan saw saw STOP) = p(the fan saw Beckham play for Real Madrid STOP) =
7 Why on earth would we want to do this?! Speech recognition was the original motivation. (Related problems are optical character recognition, handwriting recognition.)
8 Why on earth would we want to do this?! Speech recognition was the original motivation. (Related problems are optical character recognition, handwriting recognition.) The estimation techniques developed for this problem will be VERY useful for other problems in NLP
9 A Naive Method We have N training sentences For any sentence x 1... x n, c(x 1... x n ) is the number of times the sentence is seen in our training data A naive estimate: p(x 1... x n ) = c(x 1... x n ) N
10 Overview The language modeling problem Trigram models Evaluating language models: perplexity Estimation techniques: Linear interpolation Discounting methods
11 Markov Processes Consider a sequence of random variables X 1, X 2,... X n. Each random variable can take any value in a finite set V. For now we assume the length n is fixed (e.g., n = 100). Our goal: model P (X 1 = x 1, X 2 = x 2,..., X n = x n )
12 First-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n )
13 First-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n ) n = P (X 1 = x 1 ) P (X i = x i X 1 = x 1,..., X i 1 = x i 1 ) i=2
14 First-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n ) n = P (X 1 = x 1 ) P (X i = x i X 1 = x 1,..., X i 1 = x i 1 ) = P (X 1 = x 1 ) i=2 n P (X i = x i X i 1 = x i 1 ) i=2
15 First-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n ) n = P (X 1 = x 1 ) P (X i = x i X 1 = x 1,..., X i 1 = x i 1 ) = P (X 1 = x 1 ) i=2 n P (X i = x i X i 1 = x i 1 ) i=2 The first-order Markov assumption: For any i {2... n}, for any x 1... x i, P (X i = x i X 1 = x 1... X i 1 = x i 1 ) = P (X i = x i X i 1 = x i 1 )
16 Second-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n )
17 Second-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n ) = P (X 1 = x 1 ) P (X 2 = x 2 X 1 = x 1 ) n P (X i = x i X i 2 = x i 2, X i 1 = x i 1 ) i=3
18 Second-Order Markov Processes P (X 1 = x 1, X 2 = x 2,... X n = x n ) = P (X 1 = x 1 ) P (X 2 = x 2 X 1 = x 1 ) n P (X i = x i X i 2 = x i 2, X i 1 = x i 1 ) = i=3 n P (X i = x i X i 2 = x i 2, X i 1 = x i 1 ) i=1 (For convenience we assume x 0 = x 1 = *, where * is a special start symbol.)
19 Modeling Variable Length Sequences We would like the length of the sequence, n, to also be a random variable A simple solution: always define X n = STOP where STOP is a special symbol
20 Modeling Variable Length Sequences We would like the length of the sequence, n, to also be a random variable A simple solution: always define X n = STOP where STOP is a special symbol Then use a Markov process as before: = P (X 1 = x 1, X 2 = x 2,... X n = x n ) n P (X i = x i X i 2 = x i 2, X i 1 = x i 1 ) i=1 (For convenience we assume x 0 = x 1 = *, where * is a special start symbol.)
21 Trigram Language Models A trigram language model consists of: 1. A finite set V 2. A parameter q(w u, v) for each trigram u, v, w such that w V {STOP}, and u, v V {*}.
22 Trigram Language Models A trigram language model consists of: 1. A finite set V 2. A parameter q(w u, v) for each trigram u, v, w such that w V {STOP}, and u, v V {*}. For any sentence x 1... x n where x i V for i = 1... (n 1), and x n = STOP, the probability of the sentence under the trigram language model is p(x 1... x n ) = n q(x i x i 2, x i 1 ) i=1 where we define x 0 = x 1 = *.
23 An Example For the sentence the dog barks STOP we would have p(the dog barks STOP) = q(the *, *) q(dog *, the) q(barks the, dog) q(stop dog, barks)
24 The Trigram Estimation Problem Remaining estimation problem: For example: q(w i w i 2, w i 1 ) q(laughs the, dog)
25 The Trigram Estimation Problem Remaining estimation problem: For example: q(w i w i 2, w i 1 ) q(laughs the, dog) A natural estimate (the maximum likelihood estimate ): q(w i w i 2, w i 1 ) = Count(w i 2, w i 1, w i ) Count(w i 2, w i 1 ) q(laughs the, dog) = Count(the, dog, laughs) Count(the, dog)
26 Sparse Data Problems A natural estimate (the maximum likelihood estimate ): q(w i w i 2, w i 1 ) = Count(w i 2, w i 1, w i ) Count(w i 2, w i 1 ) q(laughs the, dog) = Count(the, dog, laughs) Count(the, dog) Say our vocabulary size is N = V, then there are N 3 parameters in the model. e.g., N = 20, , = parameters
27 Overview The language modeling problem Trigram models Evaluating language models: perplexity Estimation techniques: Linear interpolation Discounting methods
28 Evaluating a Language Model: Perplexity We have some test data, m sentences s 1, s 2, s 3,..., s m
29 Evaluating a Language Model: Perplexity We have some test data, m sentences s 1, s 2, s 3,..., s m We could look at the probability under our model m i=1 p(s i). Or more conveniently, the log probability m log p(s i ) = i=1 m log p(s i ) i=1
30 Evaluating a Language Model: Perplexity We have some test data, m sentences s 1, s 2, s 3,..., s m We could look at the probability under our model m i=1 p(s i). Or more conveniently, the log probability m log p(s i ) = i=1 m log p(s i ) i=1 In fact the usual evaluation measure is perplexity Perplexity = 2 l where l = 1 M m log p(s i ) i=1 and M is the total number of words in the test data.
31 Some Intuition about Perplexity Say we have a vocabulary V, and N = V + 1 and model that predicts q(w u, v) = 1 N for all w V {STOP}, for all u, v V {*}. Easy to calculate the perplexity in this case: Perplexity = 2 l where l = log 1 N Perplexity = N Perplexity is a measure of effective branching factor
32 Typical Values of Perplexity Results from Goodman ( A bit of progress in language modeling ), where V = 50, 000 A trigram model: p(x 1... x n ) = n i=1 q(x i x i 2, x i 1 ). Perplexity = 74
33 Typical Values of Perplexity Results from Goodman ( A bit of progress in language modeling ), where V = 50, 000 A trigram model: p(x 1... x n ) = n i=1 q(x i x i 2, x i 1 ). Perplexity = 74 A bigram model: p(x 1... x n ) = n i=1 q(x i x i 1 ). Perplexity = 137
34 Typical Values of Perplexity Results from Goodman ( A bit of progress in language modeling ), where V = 50, 000 A trigram model: p(x 1... x n ) = n i=1 q(x i x i 2, x i 1 ). Perplexity = 74 A bigram model: p(x 1... x n ) = n i=1 q(x i x i 1 ). Perplexity = 137 A unigram model: p(x 1... x n ) = n i=1 q(x i). Perplexity = 955
35 Some History Shannon conducted experiments on entropy of English i.e., how good are people at the perplexity game? C. Shannon. Prediction and entropy of printed English. Bell Systems Technical Journal, 30:50 64, 1951.
36 Some History Chomsky (in Syntactic Structures (1957)): Second, the notion grammatical cannot be identified with meaningful or significant in any semantic sense. Sentences (1) and (2) are equally nonsensical, but any speaker of English will recognize that only the former is grammatical. (1) Colorless green ideas sleep furiously. (2) Furiously sleep ideas green colorless Third, the notion grammatical in English cannot be identified in any way with the notion high order of statistical approximation to English. It is fair to assume that neither sentence (1) nor (2) (nor indeed any part of these sentences) has ever occurred in an English discourse. Hence, in any statistical model for grammaticalness, these sentences will be ruled out on identical grounds as equally remote from English. Yet (1), though nonsensical, is grammatical, while (2) is not....
37 Overview The language modeling problem Trigram models Evaluating language models: perplexity Estimation techniques: Linear interpolation Discounting methods
38 Sparse Data Problems A natural estimate (the maximum likelihood estimate ): q(w i w i 2, w i 1 ) = Count(w i 2, w i 1, w i ) Count(w i 2, w i 1 ) q(laughs the, dog) = Count(the, dog, laughs) Count(the, dog) Say our vocabulary size is N = V, then there are N 3 parameters in the model. e.g., N = 20, , = parameters
39 The Bias-Variance Trade-Off Trigram maximum-likelihood estimate q ML (w i w i 2, w i 1 ) = Count(w i 2, w i 1, w i ) Count(w i 2, w i 1 ) Bigram maximum-likelihood estimate q ML (w i w i 1 ) = Count(w i 1, w i ) Count(w i 1 ) Unigram maximum-likelihood estimate q ML (w i ) = Count(w i) Count()
40 Linear Interpolation Take our estimate q(w i w i 2, w i 1 ) to be q(w i w i 2, w i 1 ) = λ 1 q ML (w i w i 2, w i 1 ) +λ 2 q ML (w i w i 1 ) +λ 3 q ML (w i ) where λ 1 + λ 2 + λ 3 = 1, and λ i 0 for all i.
41 Linear Interpolation (continued) Our estimate correctly defines a distribution (define V = V {STOP}): w V q(w u, v)
42 Linear Interpolation (continued) Our estimate correctly defines a distribution (define V = V {STOP}): w V q(w u, v) = w V [λ 1 q ML (w u, v) + λ 2 q ML (w v) + λ 3 q ML (w)]
43 Linear Interpolation (continued) Our estimate correctly defines a distribution (define V = V {STOP}): w V q(w u, v) = w V [λ 1 q ML (w u, v) + λ 2 q ML (w v) + λ 3 q ML (w)] = λ 1 w q ML (w u, v) + λ 2 w q ML (w v) + λ 3 w q ML (w)
44 Linear Interpolation (continued) Our estimate correctly defines a distribution (define V = V {STOP}): w V q(w u, v) = w V [λ 1 q ML (w u, v) + λ 2 q ML (w v) + λ 3 q ML (w)] = λ 1 w q ML (w u, v) + λ 2 w q ML (w v) + λ 3 w q ML (w) = λ 1 + λ 2 + λ 3
45 Linear Interpolation (continued) Our estimate correctly defines a distribution (define V = V {STOP}): w V q(w u, v) = w V [λ 1 q ML (w u, v) + λ 2 q ML (w v) + λ 3 q ML (w)] = λ 1 w q ML (w u, v) + λ 2 w q ML (w v) + λ 3 w q ML (w) = λ 1 + λ 2 + λ 3 = 1
46 Linear Interpolation (continued) Our estimate correctly defines a distribution (define V = V {STOP}): w V q(w u, v) = w V [λ 1 q ML (w u, v) + λ 2 q ML (w v) + λ 3 q ML (w)] = λ 1 w q ML (w u, v) + λ 2 w q ML (w v) + λ 3 w q ML (w) = λ 1 + λ 2 + λ 3 = 1 (Can show also that q(w u, v) 0 for all w V )
47 How to estimate the λ values? Hold out part of training set as validation data
48 How to estimate the λ values? Hold out part of training set as validation data Define c (w 1, w 2, w 3 ) to be the number of times the trigram (w 1, w 2, w 3 ) is seen in validation set
49 How to estimate the λ values? Hold out part of training set as validation data Define c (w 1, w 2, w 3 ) to be the number of times the trigram (w 1, w 2, w 3 ) is seen in validation set Choose λ 1, λ 2, λ 3 to maximize: L(λ 1, λ 2, λ 3 ) = c (w 1, w 2, w 3 ) log q(w 3 w 1, w 2 ) w 1,w 2,w 3 such that λ 1 + λ 2 + λ 3 = 1, and λ i 0 for all i, and where q(w i w i 2, w i 1 ) = λ 1 q ML (w i w i 2, w i 1 ) +λ 2 q ML (w i w i 1 ) +λ 3 q ML (w i )
50 Allowing the λ s to vary Take a function Π that partitions histories e.g., 1 If Count(w i 1, w i 2 ) = 0 2 If 1 Count(w Π(w i 2, w i 1 ) = i 1, w i 2 ) 2 3 If 3 Count(w i 1, w i 2 ) 5 4 Otherwise Introduce a dependence of the λ s on the partition: q(w i w i 2, w i 1 ) = λ Π(w i 2,w i 1 ) 1 q ML (w i w i 2, w i 1 ) +λ Π(w i 2,w i 1 ) 2 q ML (w i w i 1 ) +λ Π(w i 2,w i 1 ) 3 q ML (w i ) where λ Π(w i 2,w i 1 ) 1 + λ Π(w i 2,w i 1 ) 2 + λ Π(w i 2,w i 1 ) 3 = 1, and λ Π(w i 2,w i 1 ) i 0 for all i.
51 Overview The language modeling problem Trigram models Evaluating language models: perplexity Estimation techniques: Linear interpolation Discounting methods
52 Discounting Methods Say we ve seen the following counts: x Count(x) q ML (w i w i 1 ) the 48 the, dog 15 15/48 the, woman 11 11/48 the, man 10 10/48 the, park 5 5/48 the, job 2 2/48 the, telescope 1 1/48 the, manual 1 1/48 the, afternoon 1 1/48 the, country 1 1/48 the, street 1 1/48 The maximum-likelihood estimates are high (particularly for low count items)
53 Discounting Methods Now define discounted counts, Count (x) = Count(x) 0.5 New estimates: x Count(x) Count (x) the 48 Count (x) Count(the) the, dog /48 the, woman /48 the, man /48 the, park /48 the, job /48 the, telescope /48 the, manual /48 the, afternoon /48 the, country /48 the, street /48
54 Discounting Methods (Continued) We now have some missing probability mass : α(w i 1 ) = 1 w Count (w i 1, w) Count(w i 1 ) e.g., in our example, α(the) = /48 = 5/48
55 Katz Back-Off Models (Bigrams) For a bigram model, define two sets A(w i 1 ) = {w : Count(w i 1, w) > 0} B(w i 1 ) = {w : Count(w i 1, w) = 0} A bigram model Count (w i 1,w i ) If w i A(w i 1 ) Count(w i 1 ) q BO (w i w i 1 ) = α(w i 1 ) q ML (w i) If w w B(w i 1 ) q ML (w) i B(w i 1 ) where α(w i 1 ) = 1 w A(w i 1 ) Count (w i 1, w) Count(w i 1 )
56 Katz Back-Off Models (Trigrams) For a trigram model, first define two sets A(w i 2, w i 1 ) = {w : Count(w i 2, w i 1, w) > 0} B(w i 2, w i 1 ) = {w : Count(w i 2, w i 1, w) = 0} A trigram model is defined in terms of the bigram model: Count (w i 2,w i 1,w i ) Count(w i 2,w i 1 ) If w i A(w i 2, w i 1 ) q BO (w i w i 2, w i 1 ) = α(w i 2,w i 1 )q BO (w i w i 1 ) w B(w i 2,w i 1 ) q BO(w w i 1 ) If w i B(w i 2, w i 1 ) where α(w i 2, w i 1 ) = 1 w A(w i 2,w i 1 ) Count (w i 2, w i 1, w) Count(w i 2, w i 1 )
57 Summary Three steps in deriving the language model probabilities: 1. Expand p(w 1, w 2... w n ) using Chain rule. 2. Make Markov Independence Assumptions p(w i w 1, w 2... w i 2, w i 1 ) = p(w i w i 2, w i 1 ) 3. Smooth the estimates using low order counts. Other methods used to improve language models: Topic or long-range features. Syntactic models. It s generally hard to improve on trigram models though!!
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