N-Grams and Language Modeling

N-grams & Language Modeling A bad language model

A bad language model

A bad language model Herman is reprinted with permission from LaughingStock Licensing Inc., Ottawa Canada. All rights reserved.

What’s a Language Model A Language model is a probability distribution over word sequences P(“And nothing but the truth”)  0.001 P(“And nuts sing on the roof”)  0

What’s a language model for? Speech recognition Handwriting recognition Spelling correction Optical character recognition Machine translation (and anyone doing statistical modeling)

Try to complete the following The quiz was ------ In this course, I want to get a good ----- Can I make a telephone ----- My friend has a fast ----- This is too ------- 4/17/2017 7

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

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.

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

Claim A useful part of the knowledge needed to allow Word Prediction can be captured using simple statistical techniques In particular, we'll rely on the notion of the probability of a sequence (a phrase, a sentence) But, Why would you want to assign a probability to a sentence or… Why would you want to predict the next word… Lots of applications

Lots of applications Example applications that employ language models: Speech recognition Handwriting recognition Spelling correction Machine translation systems Optical character recognizers 4/17/2017 12

Real Word Spelling Errors Mental confusions (cognitive) Their/they’re/there To/too/two Weather/whether Typos that result in real words Lave for Have 4/17/2017 13

Real Word Spelling Errors They are leaving in about fifteen minuets to go to her horse. The study was conducted mainly be John Black. The design an construction of the system will take more than a year. Hopefully, all with continue smoothly in my absence. I need to notified the bank of…. He is trying to fine out. horse: house, minuets: minutes be: by an: and With: will notified: notify fine: find 4/17/2017 14

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 4/17/2017 15

Applications 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 recommendó una exploración del gato.

Simple N-Grams Assume a language has V word types in its lexicon, how likely is word x to follow word y? Simplest model of word probability: 1/V Alternative 1: estimate likelihood of x occurring in new text based on its general frequency of occurrence estimated from a corpus (unigram probability) popcorn is more likely to occur than unicorn Alternative 2: condition the likelihood of x occurring in the context of previous words (bigrams, trigrams,…) mythical unicorn is more likely than mythical popcorn

N-grams A simple model of language Computes a probability for observed input. Probability is the likelihood of the observation being generated by the same source as the training data Such a model is often called a language model

Computing the Probability of a Word Sequence P(w1, …, wn) = P(w1).P(w2|w1).P(w3|w1,w2). … P(wn|w1, …,wn-1) P(the mythical unicorn) = P(the) P(mythical|the) P(unicorn|the mythical) The longer the sequence, the less likely we are to find it in a training corpus P(Most biologists and folklore specialists believe that in fact the mythical unicorn horns derived from the narwhal) Solution: approximate using n-grams

Bigram Model Approximate by P(unicorn|the mythical) by P(unicorn|mythical) Markov assumption: the probability of a word depends only on the probability of a limited history Generalization: the probability of a word depends only on the probability of the n previous words trigrams, 4-grams, … the higher n is, the more data needed to train backoff models

Using N-Grams For N-gram models  P(wn-1,wn) = P(wn | wn-1) P(wn-1) By the Chain Rule we can decompose a joint probability, e.g. P(w1,w2,w3) P(w1,w2, ...,wn) = P(w1|w2,w3,...,wn) P(w2|w3, ...,wn) … P(wn-1|wn) P(wn) For bigrams then, the probability of a sequence is just the product of the conditional probabilities of its bigrams P(the,mythical,unicorn) = P(unicorn|mythical) P(mythical|the) P(the|<start>)

The n-gram Approximation Assume each word depends only on the previous (n-1) words (n words total) For example for trigrams (3-grams): P(“the|… whole truth and nothing but”)  P(“the|nothing but”) P(“truth|… whole truth and nothing but the”)  P(“truth|but the”)

n-grams, continued How do we find probabilities? Get real text, and start counting! P(“the | nothing but”)  C(“nothing but the”) / C(“nothing but”)

Unigram probabilities (1-gram) http://www.wordcount.org/main.php Most likely to transition to “the”, least likely to transition to “conquistador”. Bigram probabilities (2-gram) Given “the” as the last word, more likely to go to “conquistador” than to “the” again.

N-grams for Language Generation C. E. Shannon, ``A mathematical theory of communication,'' Bell System Technical Journal, vol. 27, pp. 379-423 and 623-656, July and October, 1948. Unigram: 5. …Here words are chosen independently but with their appropriate frequencies. REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME CAN DIFFERENT NATURAL HERE HE THE A IN CAME THE TO OF TO EXPERT GRAY COME TO FURNISHES THE LINE MESSAGE HAD BE THESE. Bigram: 6. Second-order word approximation. The word transition probabilities are correct but no further structure is included. THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHARACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED.

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

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 don’t ? Gonna ? In Japanese and Chinese text -- how do we identify a word?

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

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

Simple N-Grams Assume a language has V word types in its lexicon, how likely is word x to follow word y? Simplest model of word probability: 1/V Alternative 1: estimate likelihood of x occurring in new text based on its general frequency of occurrence estimated from a corpus (unigram probability) popcorn is more likely to occur than unicorn Alternative 2: condition the likelihood of x occurring in the context of previous words (bigrams, trigrams,…) mythical unicorn is more likely than mythical popcorn

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

A Simple Example P(I want to each Chinese food) = P(I | <start>) P(want | I) P(to | want) P(eat | to) P(Chinese | eat) P(food | Chinese)

A Bigram Grammar Fragment from BERP .001 Eat British .03 Eat today .007 Eat dessert .04 Eat Indian .01 Eat tomorrow Eat a .02 Eat Mexican Eat at Eat Chinese .05 Eat dinner Eat in .06 Eat lunch Eat breakfast Eat some Eat Thai .16 Eat on

.01 British lunch .05 Want a British cuisine .65 Want to .15 British restaurant .04 I have .60 British food .08 I don’t .02 To be .29 I would .09 To spend .32 I want .14 To have <start> I’m .26 To eat <start> Tell Want Thai .06 <start> I’d Want some .25 <start> I

vs. I want to eat Chinese food = .00015 P(I want to eat British food) = P(I|<start>) 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

BERP Bigram Counts 1 4 Lunch 17 19 Food 120 2 Chinese 52 Eat 12 3 860 1 4 Lunch 17 19 Food 120 2 Chinese 52 Eat 12 3 860 10 To 6 8 786 Want 13 1087 I lunch

BERP Bigram Probabilities Normalization: divide each row's counts by appropriate unigram counts for wn-1 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. 459 1506 213 938 3256 1215 3437 Lunch Food Chinese Eat To Want I

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

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 ...

Approximating Shakespeare As we increase the value of N, the accuracy of the n-gram model increases, since choice of next word becomes increasingly constrained Generating sentences with random unigrams... Every enter now severally so, let Hill he late speaks; or! a more to leg less first you enter With bigrams... What means, sir. I confess she? then all sorts, he is trim, captain. Why dost stand forth thy canopy, forsooth; he is this palpable hit the King Henry.

Trigrams Quadrigrams Sweet prince, Falstaff shall die. This shall forbid it should be branded, if renown made it empty. Quadrigrams What! I will go seek the traitor Gloucester. Will you not tell me who I am?

There are 884,647 tokens, with 29,066 word form types, in about a one million word Shakespeare corpus Shakespeare produced 300,000 bigram types out of 844 million possible bigrams: so, 99.96% of the possible bigrams were never seen (have zero entries in the table) Quadrigrams worse: What's coming out looks like Shakespeare because it is Shakespeare

N-Gram Training Sensitivity If we repeated the Shakespeare experiment but trained our n-grams on a Wall Street Journal corpus, what would we get? This has major implications for corpus selection or design

Some Useful Empirical Observations 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 Some of the zeroes in the table are really zeros But others are simply low frequency events you haven't seen yet. How to address?

Unknown words Unknown or Out of vocabulary (OOV) words Open Vocabulary system – model the unknown word by <UNK> Training is as follows: Choose a vocabulary Convert any word in training set not belonging to this set to <UNK> Estimate the probabilities for <UNK> from its counts

Smoothing Techniques Every n-gram training matrix is sparse, even for very large corpora (Zipf’s 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?

Smoothing: None Called Maximum Likelihood estimate. Terrible on test data: If no occurrences of C(xyz), probability is 0.

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 Normalize by N to get the new unigram probability: For bigrams: Add 1 to every bigram c(wn-1 wn) + 1 Incr unigram count by vocabulary size c(wn-1) + V

Effect on BERP bigram counts

Add-one bigram probabilities

The problem

The problem Add-one has a huge effect on probabilities: e.g., P(to|want) went from .65 to .28! Too much probability gets ‘removed’ from n-grams actually encountered (more precisely: the ‘discount factor’

But this changes counts drastically: Discount: ratio of new counts to old (e.g. add-one smoothing changes the BERP bigram (to|want) from 786 to 331 (dc=.42) and p(to|want) from .65 to .28) But this changes counts drastically: too much weight given to unseen ngrams in practice, unsmoothed bigrams often work better!

Smoothing Add one smoothing: Works very badly. Add delta smoothing: Still very bad. [based on slides by Joshua Goodman]