N-Grams and Corpus Linguistics - University of Delaware

N-Grams and Corpus Linguistics Lecture #4 September 6, 2012 1 Transition Up to this point weve mostly been discussing words in isolation Now were switching to sequences of words And were going to worry about assigning probabilities to sequences of words 2 Who Cares? Why would you want to assign a probability to a sentence or

Why would you want to predict the next word Lots of applications 3 Real-Word Spelling Errors Mental confusions Their/theyre/there To/too/two Weather/whether Peace/piece Youre/your

Typos that result in real words Lave for Have 4 Real Word Spelling Errors Collect a set of common pairs of confusions Whenever a member of this set is encountered compute the probability of the sentence in which it appears Substitute the other possibilities and compute the probability of the resulting sentence Choose the higher one 5 Next Word Prediction From a NY Times story...

Stocks ... Stocks plunged this . Stocks plunged this morning, despite a cut in interest rates Stocks plunged this morning, despite a cut in interest rates by the Federal Reserve, as Wall ... Stocks plunged this morning, despite a cut in interest rates by the Federal Reserve, as Wall Street began 6 Stocks plunged this morning, despite a cut in interest rates by the Federal Reserve, as Wall Street began trading for the first time since last Stocks plunged this morning, despite a cut in interest rates by the Federal Reserve, as Wall Street began trading for the

first time since last Tuesday's terrorist attacks. 7 Human Word Prediction Clearly, at least some of us have the ability to predict future words in an utterance. How? Domain knowledge Syntactic knowledge Lexical knowledge 8 Word Prediction Guess the next word... ... I notice three guys standing on the ??? There are many sources of knowledge

that can be used to inform this task, including arbitrary world knowledge. But it turns out that you can do pretty well by simply looking at the preceding words and keeping track of some fairly simple counts. 02/24/20 Speech and Language Processing - Jurafsky and Martin 9 Word Prediction We can formalize this task using what are called N-gram models. N-grams are token sequences of length N. Our earlier example contains the following 2-grams (aka bigrams) (I notice), (notice three), (three guys), (guys

standing), (standing on), (on the) Given knowledge of counts of N-grams such as these, we can guess likely next words in a sequence. 02/24/20 Speech and Language Processing - Jurafsky and Martin 10 N-Gram Models More formally, we can use knowledge of the counts of N-grams to assess the conditional probability of candidate words as the next word in a sequence. Or, we can use them to assess the probability of an entire sequence of

words. Pretty much the same thing as well see... 02/24/20 Speech and Language Processing - Jurafsky and Martin 11 Applications Why do we want to predict a word, given some preceding words? Rank the likelihood of sequences containing various alternative hypotheses, e.g. for ASR Theatre owners say popcorn/unicorn sales have doubled... Assess the likelihood/goodness of a sentence, e.g. for text generation or machine translation The doctor recommended a cat scan. El doctor recommend una exploracin del gato.

12 N-Gram Models of Language Use the previous N-1 words in a sequence to predict the next word Language Model (LM) unigrams, bigrams, trigrams, How do we train these models? Very large corpora 13 Counting Words in Corpora What is a word?

e.g., are cat and cats the same word? September and Sept? zero and oh? Is _ a word? * ? ( ? How many words are there in dont ? Gonna ? In Japanese and Chinese text -- how do we identify a word? 14 Terminology Sentence: unit of written language Utterance: unit of spoken language Word Form: the inflected form that appears in the corpus Lemma: an abstract form, shared by word forms

having the same stem, part of speech, and word sense Types: number of distinct words in a corpus (vocabulary size) Tokens: total number of words 15 Counting: Corpora So what happens when we look at large bodies of text instead of single utterances? Brown et al (1992) large corpus of English text 583 million wordform tokens 293,181 wordform types Google Crawl of 1,024,908,267,229 English tokens 13,588,391 wordform types That seems like a lot of types... After all, even large dictionaries of English

have only around 500k types. Why so many here? Numbers Misspellings Names Acronyms etc 02/24/20 Speech and Language Processing - Jurafsky and Martin 16 Corpora Corpora are online collections of text and speech

Brown Corpus Wall Street Journal AP news Hansards DARPA/NIST text/speech corpora (Call Home, ATIS, switchboard, Broadcast News, TDT, Communicator) TRAINS, Radio News 17 Language Modeling Back to word prediction We can model the word prediction task as the ability to assess the conditional probability of a word given the previous words in the sequence

P(wn|w1,w2wn-1) Well call a statistical model that can assess this a Language Model 02/24/20 Speech and Language Processing - Jurafsky and Martin 18 Language Modeling How might we go about calculating such a conditional probability? One way is to use the definition of conditional probabilities and look for counts. So to get P(the | its water is so transparent that)

By definition thats P(its water is so transparent that the) P(its water is so transparent that) We can get each of those from counts in a large corpus. 02/24/20 Speech and Language Processing - Jurafsky and Martin 19 Very Easy Estimate How to estimate? P(the | its water is so transparent that) P(the | its water is so transparent that) = Count(its water is so transparent that the) Count(its water is so transparent that)

02/24/20 Speech and Language Processing - Jurafsky and Martin 20 Very Easy Estimate According to Google those counts are 5/9. Unfortunately... 2 of those were to these slides... So maybe its really 3/7 In any case, thats not terribly convincing due to the small numbers involved. 02/24/20 Speech and Language Processing - Jurafsky and Martin

21 Language Modeling Unfortunately, for most sequences and for most text collections we wont get good estimates from this method. What were likely to get is 0. Or worse 0/0. Clearly, well have to be a little more clever. Lets use the chain rule of probability And a particularly useful independence assumption. 02/24/20 Speech and Language Processing - Jurafsky and Martin

22 The Chain Rule Recall the definition of conditional probabilities P ( A^ B ) P( A | B) P( B) Rewriting: P( A^ B) P( A | B) P( B) For sequences... P(A,B,C,D) = P(A)P(B|A)P(C|A,B)P(D|A,B,C) In general P(x1,x2,x3,xn) = P(x1)P(x2|x1)P(x3|x1,x2)P(xn| x1xn-1) 02/24/20

Speech and Language Processing - Jurafsky and Martin 23 The Chain Rule P(its water was so transparent)= P(its)* P(water|its)* P(was|its water)* P(so|its water was)* P(transparent|its water was so) 02/24/20 Speech and Language Processing - Jurafsky and Martin 24

Example The big red dog P(The)*P(big|the)*P(red|the big)*P(dog|the big red) Better P(The| ) written as P(The | ) 25 General Case The word sequence from position 1 to n is So the probability of a sequence is

w1 n P ( w1n ) P( w1) P( w2 | w1) P( w3 | w12 )...P ( wn | w1n 1 ) n P( w1) k 2 P( wk | w1k 1 ) 26 Unfortunately That doesnt help since its unlikely well ever gather the right statistics for the prefixes. 27 Markov Assumption

Assume that the entire prefix history isnt necessary. In other words, an event doesnt depend on all of its history, just a fixed length near history 28 Markov Assumption So for each component in the product replace each with the approximation (assuming a prefix of N) n 1 1 P ( wn | w n 1 n N 1 ) P ( wn | w

) 29 N-Grams The big red dog Unigrams: Bigrams: Trigrams: Four-grams: P(dog) P(dog|red)

P(dog|big red) P(dog|the big red) In general, well be dealing with P(Word| Some fixed prefix) 30 Caveat The formulation P(Word| Some fixed prefix) is not really appropriate in many applications. It is if were dealing with real time speech where we only have access to prefixes. But if were dealing with text we already have the right and left contexts. Theres no a priori reason to stick to left contexts. 31

Training and Testing N-Gram probabilities come from a training corpus overly narrow corpus: probabilities don't generalize overly general corpus: probabilities don't reflect task or domain A separate test corpus is used to evaluate the model, typically using standard metrics held out test set; development test set cross validation results tested for statistical significance 32 A Simple Example P(I want to eat Chinese food) = P(I | ) P(want | I) P(to | want) P(eat | to) P(Chinese | eat) P(food | Chinese) 33

A Bigram Grammar Fragment from BERP eat on .16 eat Thai .03 eat some .06 eat breakfast .03

eat lunch .06 eat in .02 eat dinner .05 eat Chinese .02 eat at .04

eat Mexican .02 eat a .04 eat tomorrow .01 eat Indian .04 eat dessert

.007 eat today .03 eat British .001 34 I .25 want some .04

Id .06 want Thai .01 Tell .04 to eat .26 Im .02

to have .14 I want .32 to spend .09 I would .29 to be

.02 I dont .08 British food .60 I have .04 British restaurant .15 want to

.65 British cuisine .01 want a .05 British lunch .01 35 P(I want to eat British food) = P(I|) P(want|I) P(to|want) P(eat|to) P(British|eat) P(food|British)

= .25*.32*.65*.26*.001*.60 = .000080 vs. I want to eat Chinese food = .00015 Probabilities seem to capture ``syntactic'' facts, ``world knowledge'' eat is often followed by an NP British food is not too popular N-gram models can be trained by counting and normalization 36 An Aside on Logs You dont really do all those multiplies. The numbers are too small and lead to underflows Convert the probabilities to logs and then do additions. To get the real probability (if you need it) go back to the antilog.

37 How do we get the N-gram probabilities? N-gram models can be trained by counting and normalization 38 Estimating Bigram Probabilities The Maximum Likelihood Estimate (MLE) count(w i 1,w i ) P(w i | w i 1 ) count(w i 1 )

02/24/20 Speech and Language Processing - Jurafsky and Martin 39 An Example I am Sam Sam I am I do not like green eggs and ham 02/24/20 Speech and Language Processing - Jurafsky and Martin 40 Maximum Likelihood Estimates The maximum likelihood estimate of some

parameter of a model M from a training set T Is the estimate that maximizes the likelihood of the training set T given the model M Suppose the word Chinese occurs 400 times in a corpus of a million words (Brown corpus) What is the probability that a random word from some other text from the same distribution will be Chinese MLE estimate is 400/1000000 = .004 This may be a bad estimate for some other corpus But it is the estimate that makes it most likely that Chinese will occur 400 times in a million word corpus. 02/24/20 Speech and Language Processing - Jurafsky and Martin

41 Berkeley Restaurant Project Sentences can you tell me about any good cantonese restaurants close by mid priced thai food is what im looking for tell me about chez panisse can you give me a listing of the kinds of food that are available im looking for a good place to eat breakfast when is caffe venezia open during the day 02/24/20 Speech and Language Processing - Jurafsky and Martin 42

BERP Bigram Counts I want to eat Chinese food lunch I 8

1087 0 13 0 0 0 want 3 0 786

0 6 8 6 to 3 0 10 860

3 0 12 eat 0 0 2 0 19 2

52 Chinese 2 0 0 0 0 120 1

food 19 0 17 0 0 0 0 lunch 4

0 0 0 0 1 0 43 BERP Bigram Probabilities Normalization: divide each row's counts by appropriate unigram counts for wn-1 I

want to eat Chinese food lunch 3437 1215 3256

938 213 1506 459 Computing the bigram probability of I I C(I,I)/C(all I) p (I|I) = 8 / 3437 = .0023 Maximum Likelihood Estimation (MLE): relative frequency of e.g. freq(w1, w2) freq(w1) 44 BERP Table: Bigram Probabilities

45 What do we learn about the language? What's being captured with ... P(want | I) = .32 P(to | want) = .65 P(eat | to) = .26 P(food | Chinese) = .56 P(lunch | eat) = .055 What about...

P(I | I) = .0023 P(I | want) = .0025 P(I | food) = .013 46 P(I | I) = .0023 I I I I want P(I | want) = .0025 I want I want P(I | food) = .013 the kind of food I want is ... 47 Kinds of Knowledge As crude as they are, N-gram probabilities capture a range of interesting facts about language. P(english|want) World knowledge = .0011

P(chinese|want) = .0065 Syntax P(to|want) = .66 P(eat | to) = .28 P(food | to) = 0 Discourse P(want | spend) = 0 P (i | ) = .25 02/24/20 Speech and Language Processing - Jurafsky and Martin 48 Shannons Method Assigning probabilities to sentences is all well and good, but its not terribly illuminating . A more

interesting task is to turn the model around and use it to generate random sentences that are like the sentences from which the model was derived. Generally attributed to Claude Shannon. 02/24/20 Speech and Language Processing - Jurafsky and Martin 49 Shannons Method Sample a random bigram (, w) according to its probability Now sample a random bigram (w, x) according to its probability Where the prefix w matches the suffix of the first.

And so on until we randomly choose a (y, ) Then string the words together I I want want to to eat eat Chinese Chinese food food 02/24/20 Speech and Language Processing - Jurafsky and Martin

50 Shakespeare 02/24/20 Speech and Language Processing - Jurafsky and Martin 51 Shakespeare as a Corpus N=884,647 tokens, V=29,066 Shakespeare produced 300,000 bigram types out of V2= 844 million possible bigrams... So, 99.96% of the possible bigrams were never seen (have zero entries in the table) This is the biggest problem in language

modeling; well come back to it. Quadrigrams are worse: What's coming out looks like Shakespeare because it is Shakespeare 02/24/20 Speech and Language Processing - Jurafsky and Martin 52 The Wall Street Journal is Not Shakespeare 02/24/20 Speech and Language Processing - Jurafsky and Martin 53

Evaluation How do we know if our models are any good? And in particular, how do we know if one model is better than another. Well Shannons game gives us an intuition. The generated texts from the higher order models sure look better. That is, they sound more like the text the model was obtained from. But what does that mean? Can we make that notion operational? 02/24/20 Speech and Language Processing - Jurafsky and Martin

54 Evaluation Standard method Train parameters of our model on a training set. Look at the models performance on some new data This is exactly what happens in the real world; we want to know how our model performs on data we havent seen So use a test set. A dataset which is different than our training set, but is drawn from the same source Then we need an evaluation metric to tell us how well our model is doing on the test set. One such metric is perplexity (to be introduced below) 02/24/20

Speech and Language Processing - Jurafsky and Martin 55 Unknown Words But once we start looking at test data, well run into words that we havent seen before (pretty much regardless of how much training data you have. With an Open Vocabulary task Create an unknown word token Training of probabilities Create a fixed lexicon L, of size V From a dictionary or A subset of terms from the training set At text normalization phase, any training word not in L changed to

Now we count that like a normal word At test time Use UNK counts for any word not in training 02/24/20 Speech and Language Processing - Jurafsky and Martin 56 Perplexity Perplexity is the probability of the test set (assigned by the language model), normalized by the number of words: Chain rule: For bigrams:

Minimizing perplexity is the same as maximizing probability The best language model is one that best predicts an unseen test set 02/24/20 Speech and Language Processing - Jurafsky and Martin 57 Lower perplexity means a better model Training 38 million words, test 1.5 million words, WSJ 02/24/20 Speech and Language Processing - Jurafsky and Martin

58 Evaluating N-Gram Models Best evaluation for a language model Put model A into an application For example, a speech recognizer Evaluate the performance of the application with model A Put model B into the application and evaluate Compare performance of the application with the two models Extrinsic evaluation 02/24/20 Speech and Language Processing - Jurafsky and Martin

59 Difficulty of extrinsic (in-vivo) evaluation of N-gram models Extrinsic evaluation This is really time-consuming Can take days to run an experiment So As a temporary solution, in order to run experiments To evaluate N-grams we often use an intrinsic evaluation, an approximation called perplexity But perplexity is a poor approximation unless the test data looks just like the training data So is generally only useful in pilot experiments (generally is not sufficient to publish) But is helpful to think about.

02/24/20 Speech and Language Processing - Jurafsky and Martin 60 Zero Counts Back to Shakespeare Recall that Shakespeare produced 300,000 bigram types out of V2= 844 million possible bigrams... So, 99.96% of the possible bigrams were never seen (have zero entries in the table) Does that mean that any sentence that contains one of those bigrams should have a probability of 0? 02/24/20

Speech and Language Processing - Jurafsky and Martin 61 Zero Counts Some of those zeros are really zeros... Things that really cant or shouldnt happen. On the other hand, some of them are just rare events. If the training corpus had been a little bigger they would have had a count (probably a count of 1!). Zipfs Law (long tail phenomenon): A small number of events occur with high frequency

A large number of events occur with low frequency You can quickly collect statistics on the high frequency events You might have to wait an arbitrarily long time to get valid statistics on low frequency events Result: Our estimates are sparse! We have no counts at all for the vast bulk of things we want to estimate! Answer: Estimate the likelihood of unseen (zero count) N-grams! 02/24/20 Speech and Language Processing - Jurafsky and Martin 62 Smoothing Techniques

Every n-gram training matrix is sparse, even for very large corpora (Zipfs law) Solution: estimate the likelihood of unseen n-grams Problems: how do you adjust the rest of the corpus to accommodate these phantom n-grams? 63 Problem Lets assume were using N-grams How can we assign a probability to a sequence where one of the component n-grams has a value of zero Assume all the words are known and have been seen Go to a lower order n-gram Back off from bigrams to unigrams Replace the zero with something else 64

Add-One (Laplace) Make the zero counts 1. Rationale: Theyre just events you havent seen yet. If you had seen them, chances are you would only have seen them once so make the count equal to 1. 65 Add-one Smoothing For unigrams: Add 1 to every word (type) count Normalize by N (tokens) /(N (tokens) +V (types)) Smoothed count (adjusted for additions to N) is

c 1 N N V i Normalize by N to get the new unigram probability: p* c 1 i N V i For bigrams: Add 1 to every bigram c(wn-1 wn) + 1 Incr unigram count by vocabulary size c(wn-1) + V 66 Original BERP Counts

67 BERP Table: Bigram Probabilities 68 BERP After Add-One Was .65 69 Add-One Smoothed BERP Reconstituted 70 Discount: ratio of new counts to old (e.g. add-one smoothing changes the BERP count (to|want) from 786 to 331 (dc=.42) and p(to|want) from .65 to .28)

Problem: add one smoothing changes counts drastically: too much weight given to unseen ngrams in practice, unsmoothed bigrams often work better! 71 Better Smoothing Intuition used by many smoothing algorithms Good-Turing Kneser-Ney Witten-Bell Is to use the count of things weve seen once to help estimate the count of things weve never seen 02/24/20

Speech and Language Processing - Jurafsky and Martin 72 Good-Turing Josh Goodman Intuition Imagine you are fishing There are 8 species: carp, perch, whitefish, trout, salmon, eel, catfish, bass You have caught 10 carp, 3 perch, 2 whitefish, 1 trout, 1 salmon, 1 eel = 18 fish How likely is it that the next fish caught is from a new species (one not seen in our

previous catch)? 3/18 Assuming so, how likely is it that next species is trout? Must be less than 1/18 02/24/20 Slide adapted from Josh Goodman Speech and Language Processing - Jurafsky and Martin 73 Good-Turing Notation: Nx is the frequency-of-frequency-x So N10=1 Number of fish species seen 10 times is 1 (carp)

N1=3 Number of fish species seen 1 is 3 (trout, salmon, eel) To estimate total number of unseen species Use number of species (words) weve seen once c0* =c1 p0 = N1/N All other estimates are adjusted (down) to give probabilities for unseen 02/24/20 Slide from Josh Goodman Speech and Language Processing - Jurafsky and Martin 74

Good-Turing Intuition Notation: Nx is the frequency-of-frequency-x So N10=1, N1=3, etc To estimate total number of unseen species Use number of species (words) weve seen once c0* =c1 p0 = N1/N p0=N1/N=3/18 All other estimates are adjusted (down) to give probabilities for unseen P(eel) = c*(1) = (1+1) 1/ 3 = 2/3 02/24/20 Slide from Josh Goodman Speech and Language Processing - Jurafsky and Martin

75 GT Fish Example 02/24/20 Speech and Language Processing - Jurafsky and Martin 76 Bigram Frequencies of Frequencies and GT Re-estimates 02/24/20 Speech and Language Processing - Jurafsky and Martin

77 Complications In practice, assume large counts (c>k for some k) are reliable: That complicates c*, making it: Also: we assume singleton counts c=1 are unreliable, so treat N-grams with count of 1 as if they were count=0 Also, need the Nk to be non-zero, so we need to smooth (interpolate) the Nk counts before computing c* from them 02/24/20 Speech and Language Processing - Jurafsky and Martin 78 Backoff and Interpolation

Another really useful source of knowledge If we are estimating: trigram p(z|x,y) but count(xyz) is zero Use info from: Bigram p(z|y) Or even: Unigram p(z) How to combine this trigram, bigram, unigram info in a valid fashion? 02/24/20 Speech and Language Processing - Jurafsky and Martin 79

Backoff Vs. Interpolation Backoff: use trigram if you have it, otherwise bigram, otherwise unigram Interpolation: mix all three 02/24/20 Speech and Language Processing - Jurafsky and Martin 80 Interpolation Simple interpolation Lambdas conditional on context: 02/24/20

Speech and Language Processing - Jurafsky and Martin 81 How to Set the Lambdas? Use a held-out, or development, corpus Choose lambdas which maximize the probability of some held-out data I.e. fix the N-gram probabilities Then search for lambda values That when plugged into previous equation Give largest probability for held-out set Can use EM to do this search 02/24/20 Speech and Language Processing - Jurafsky and Martin

82 Katz Backoff 02/24/20 Speech and Language Processing - Jurafsky and Martin 83 Why discounts P* and alpha? MLE probabilities sum to 1 So if we used MLE probabilities but backed off to lower order model when MLE prob is zero We would be adding extra probability mass And total probability would be greater than 1

02/24/20 Speech and Language Processing - Jurafsky and Martin 84 GT Smoothed Bigram Probabilities 02/24/20 Speech and Language Processing - Jurafsky and Martin 85 Intuition of Backoff+Discounting How much probability to assign to all

the zero trigrams? Use GT or other discounting algorithm to tell us How to divide that probability mass among different contexts? Use the N-1 gram estimates to tell us What do we do for the unigram words not seen in training? Out Of Vocabulary = OOV words 02/24/20 Speech and Language Processing - Jurafsky and Martin 86 OOV words: word Out Of Vocabulary = OOV words

We dont use GT smoothing for these Because GT assumes we know the number of unseen events Instead: create an unknown word token Training of probabilities Create a fixed lexicon L of size V At text normalization phase, any training word not in L changed to Now we train its probabilities like a normal word At decoding time If text input: Use UNK probabilities for any word not in training 02/24/20 Speech and Language Processing - Jurafsky and Martin

87 Practical Issues We do everything in log space Avoid underflow (also adding is faster than multiplying) 02/24/20 Speech and Language Processing - Jurafsky and Martin 88 Google N-Gram Release 02/24/20 Speech and Language Processing - Jurafsky and Martin

89 Google N-Gram Release 02/24/20 serve serve serve serve

serve serve serve serve serve serve as as as as as as as as as as the

the the the the the the the the the incoming 92 incubator 99 independent 794 index 223 indication 72 indicator 120 indicators 45 indispensable 111 indispensible 40

individual 234 Speech and Language Processing - Jurafsky and Martin 90 Google Caveat Remember the lesson about test sets and training sets... Test sets should be similar to the training set (drawn from the same distribution) for the probabilities to be meaningful. So... The Google corpus is fine if your application deals with arbitrary English text on the Web. If not then a smaller domain specific corpus is likely to yield better results. 02/24/20

Speech and Language Processing - Jurafsky and Martin 91 Summary N-gram probabilities can be used to estimate the likelihood Of a word occurring in a context (N-1) Of a sentence occurring at all Smoothing techniques deal with problems of unseen words in a corpus 92

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