�Language Modeling

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Introduction to N-grams

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Dan Jurafsky

Probabilistic Language Models

• Today’s goal: assign a probability to a sentence
• Machine Translation:
• P(high winds tonite) > P(large winds tonite)
• Spell Correction
• The office is about fifteen minuets from my house
• P(about fifteen minutes from) > P(about fifteen minuets from)
• Speech Recognition
• P(I saw a van) >> P(eyes awe of an)
• + Summarization, question-answering, etc., etc.!!

Why?

Dan Jurafsky

Probabilistic Language Modeling

• Goal: compute the probability of a sentence or sequence of words:

P(W) = P(w₁,w₂,w₃,w₄,w₅…w_n)

• Related task: probability of an upcoming word:

P(w₅|w₁,w₂,w₃,w₄)

• A model that computes either of these:

P(W) or P(w_n|w₁,w₂…w_n-1) is called a language model.

• Better: the grammar But language model or LM is standard

Dan Jurafsky

How to compute P(W)

• How to compute this joint probability:

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• P(its, water, is, so, transparent, that)

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• Intuition: let’s rely on the Chain Rule of Probability

Dan Jurafsky

Reminder: The Chain Rule

• Recall the definition of conditional probabilities

p(B|A) = P(A,B)/P(A) Rewriting: P(A,B) = P(A)P(B|A)

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• More variables:

P(A,B,C,D) = P(A)P(B|A)P(C|A,B)P(D|A,B,C)

• The Chain Rule in General

P(x₁,x₂,x₃,…,x_n) = P(x₁)P(x₂|x₁)P(x₃|x₁,x₂)…P(x_n|x₁,…,x_n-1)

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Dan Jurafsky

The Chain Rule applied to compute joint probability of words in sentence

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P(“its water is so transparent”) =

P(its) × P(water|its) × P(is|its water)

× P(so|its water is) × P(transparent|its water is so)

Dan Jurafsky

How to estimate these probabilities

• Could we just count and divide?

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• No! Too many possible sentences!
• We’ll never see enough data for estimating these

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Dan Jurafsky

Markov Assumption

• Simplifying assumption:

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• Or maybe

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Andrei Markov

Dan Jurafsky

Markov Assumption

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• In other words, we approximate each component in the product

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Dan Jurafsky

Simplest case: Unigram model

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fifth, an, of, futures, the, an, incorporated, a, a, the, inflation, most, dollars, quarter, in, is, mass

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thrift, did, eighty, said, hard, 'm, july, bullish

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that, or, limited, the

Some automatically generated sentences from a unigram model

Dan Jurafsky

Bigram model

• Condition on the previous word:

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texaco, rose, one, in, this, issue, is, pursuing, growth, in, a, boiler, house, said, mr., gurria, mexico, 's, motion, control, proposal, without, permission, from, five, hundred, fifty, five, yen

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outside, new, car, parking, lot, of, the, agreement, reached

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this, would, be, a, record, november

Dan Jurafsky

N-gram models

• We can extend to trigrams, 4-grams, 5-grams
• In general this is an insufficient model of language
• because language has long-distance dependencies:

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“The computer which I had just put into the machine room on the fifth floor crashed.”

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• But we can often get away with N-gram models

Dan Jurafsky

�Language Modeling

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Introduction to N-grams

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Dan Jurafsky

�Language Modeling

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Estimating N-gram Probabilities

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Dan Jurafsky

Estimating bigram probabilities

• The Maximum Likelihood Estimate

Dan Jurafsky

An example

<s> I am Sam </s>

<s> Sam I am </s>

<s> I do not like green eggs and ham </s>

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Dan Jurafsky

More examples: �Berkeley Restaurant Project sentences

• can you tell me about any good cantonese restaurants close by
• mid priced thai food is what i’m looking for
• tell me about chez panisse
• can you give me a listing of the kinds of food that are available
• i’m looking for a good place to eat breakfast
• when is caffe venezia open during the day

Dan Jurafsky

Raw bigram counts

• Out of 9222 sentences

Dan Jurafsky

Raw bigram probabilities

• Normalize by unigrams:

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• Result:

Dan Jurafsky

Bigram estimates of sentence probabilities

P(<s> I want english food </s>) =

P(I|<s>)

× P(want|I)

× P(english|want)

× P(food|english)

× P(</s>|food)

= .000031

Dan Jurafsky

What kinds of knowledge?

• P(english|want) = .0011
• P(chinese|want) = .0065
• P(to|want) = .66
• P(eat | to) = .28
• P(food | to) = 0
• P(want | spend) = 0
• P (i | <s>) = .25

Dan Jurafsky

Practical Issues

• We do everything in log space
• Avoid underflow
• (also adding is faster than multiplying)

Dan Jurafsky

Language Modeling Toolkits

• SRILM
• http://www.speech.sri.com/projects/srilm/
• KenLM
• https://kheafield.com/code/kenlm/

Dan Jurafsky

Google N-Gram Release, August 2006

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Dan Jurafsky

Google N-Gram Release

• serve as the incoming 92
• serve as the incubator 99
• serve as the independent 794
• serve as the index 223
• serve as the indication 72
• serve as the indicator 120
• serve as the indicators 45
• serve as the indispensable 111
• serve as the indispensible 40
• serve as the individual 234

http://googleresearch.blogspot.com/2006/08/all-our-n-gram-are-belong-to-you.html

Dan Jurafsky

Google Book N-grams

• http://ngrams.googlelabs.com/

Dan Jurafsky

�Language Modeling

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Estimating N-gram Probabilities

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Dan Jurafsky

�Language Modeling

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Evaluation and Perplexity

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Dan Jurafsky

Evaluation: How good is our model?

• Does our language model prefer good sentences to bad ones?
• Assign higher probability to “real” or “frequently observed” sentences
• Than “ungrammatical” or “rarely observed” sentences?
• We train parameters of our model on a training set.
• We test the model’s performance on data we haven’t seen.
• A test set is an unseen dataset that is different from our training set, totally unused.
• An evaluation metric tells us how well our model does on the test set.

Dan Jurafsky

(Extra Slide not in video) �Training on the test set

• We can’t allow test sentences into the training set
• We will assign it an artificially high probability when we set it in the test set
• “Training on the test set”
• Bad science!
• And violates the honor code

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Dan Jurafsky

Extrinsic evaluation of N-gram models

• Best evaluation for comparing models A and B
• Put each model in a task
• spelling corrector, speech recognizer, MT system
• Run the task, get an accuracy for A and for B
• How many misspelled words corrected properly
• How many words translated correctly
• Compare accuracy for A and B

Dan Jurafsky

Difficulty of extrinsic (in-vivo) evaluation of N-gram models

• Extrinsic evaluation
• Time-consuming; can take days or weeks
• So
• Sometimes use intrinsic evaluation: perplexity
• Bad approximation
• unless the test data looks just like the training data
• So generally only useful in pilot experiments
• But is helpful to think about.

Dan Jurafsky

Intuition of Perplexity

• The Shannon Game:
• How well can we predict the next word?

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• Unigrams are terrible at this game. (Why?)
• A better model of a text
• is one which assigns a higher probability to the word that actually occurs

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I always order pizza with cheese and ____

The 33^rd President of the US was ____

I saw a ____

mushrooms 0.1

pepperoni 0.1

anchovies 0.01

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fried rice 0.0001

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and 1e-100

Dan Jurafsky

Perplexity

Perplexity is the inverse probability of the test set, normalized by the number of words:

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Chain rule:

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For bigrams:

Minimizing perplexity is the same as maximizing probability

The best language model is one that best predicts an unseen test set

• Gives the highest P(sentence)

Dan Jurafsky

The Shannon Game intuition for perplexity

• From Josh Goodman
• How hard is the task of recognizing digits ‘0,1,2,3,4,5,6,7,8,9’
• Perplexity 10
• How hard is recognizing (30,000) names at Microsoft.
• Perplexity = 30,000
• If a system has to recognize
• Operator (1 in 4)
• Sales (1 in 4)
• Technical Support (1 in 4)
• 30,000 names (1 in 120,000 each)
• Perplexity is 52.6
• Perplexity is weighted equivalent branching factor

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Dan Jurafsky

The Shannon Game intuition for perplexity

A call-routing phone system gets 120K calls and has to recognize

• "Operator" (let's say this occurs 1 in 4 calls)
• "Sales" (1 in 4)
• "Technical Support" (1 in 4)
• 30,000 different names (each name occurring 1 time in the 120K calls)

To get the perplexity of this sequence of length 120K:

1) multiply 120K probabilities (90K of which are 1/4 and 30K of which are 1/120K)

2) take the inverse 120,000th root:

Perp = (¼ * ¼ * ¼* ¼ * ¼ * …. * 1/120K * 1/120K * ….)^(-1/120K)

Can be arithmetically simplified to just N = 4:  operator (1/4), sales (1/4), tech support (1/4), and 30,000 names (1/120,000):

Perplexity= (¼ * ¼ * ¼ * 1/120K)^(-1/4) = 52.6

Perplexity as branching factor

• Let’s suppose a sentence consisting of random digits
• What is the perplexity of this sentence according to a model that assign P=1/10 to each digit?

Dan Jurafsky

Lower perplexity = better model

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• Training 38 million words, test 1.5 million words, WSJ

Dan Jurafsky

�Language Modeling

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Evaluation and Perplexity

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Dan Jurafsky

�Language Modeling

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Generalization and zeros

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Dan Jurafsky

The Shannon Visualization Method

• Choose a random bigram

(<s>, w) according to its probability

• Now choose a random bigram (w, x) according to its probability
• And so on until we choose </s>
• Then string the words together

<s> I

I want

want to

to eat

eat Chinese

Chinese food

food </s>

I want to eat Chinese food

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Dan Jurafsky

Approximating Shakespeare

Dan Jurafsky

Shakespeare as 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)
• Quadrigrams worse: What's coming out looks like Shakespeare because it is Shakespeare

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Dan Jurafsky

The wall street journal is not shakespeare (no offense)

Dan Jurafsky

Can you guess the author of these random 3-gram sentences?

• They also point to ninety nine point six billion dollars from two hundred four oh six three percent of the rates of interest stores as Mexico and gram Brazil on market conditions
• This shall forbid it should be branded, if renown made it empty.
• “You are uniformly charming!” cried he, with a smile of associating and now and then I bowed and they perceived a chaise and four to wish for.

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Dan Jurafsky

The perils of overfitting

• N-grams only work well for word prediction if the test corpus looks like the training corpus
• In real life, it often doesn’t
• We need to train robust models that generalize!
• One kind of generalization: Zeros!
• Things that don’t ever occur in the training set
• But occur in the test set

Dan Jurafsky

Zeros

• Training set:

… denied the allegations

… denied the reports

… denied the claims

… denied the request

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P(“offer” | denied the) = 0

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• Test set

… denied the offer

… denied the loan

Dan Jurafsky

Zero probability bigrams

• Bigrams with zero probability
• mean that we will assign 0 probability to the test set!
• And hence we cannot compute perplexity (can’t divide by 0)!

Dan Jurafsky

�Language Modeling

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Generalization and zeros

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Dan Jurafsky

�Language Modeling

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Smoothing: Add-one (Laplace) smoothing

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Dan Jurafsky

The intuition of smoothing (from Dan Klein)

• When we have sparse statistics:

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• Steal probability mass to generalize better

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P(w | denied the)

3 allegations

2 reports

1 claims

1 request

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7 total

P(w | denied the)

2.5 allegations

1.5 reports

0.5 claims

0.5 request

2 other

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7 total

allegations

reports

claims

attack

request

man

outcome

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allegations

attack

man

outcome

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allegations

reports

claims

request

Dan Jurafsky

Add-one estimation

• Also called Laplace smoothing
• Pretend we saw each word one more time than we did
• Just add one to all the counts!

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• MLE estimate:

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• Add-1 estimate:

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Dan Jurafsky

Maximum Likelihood Estimates

• The maximum likelihood estimate
• of some parameter of a model M from a training set T
• maximizes the likelihood of the training set T given the model M
• Suppose the word “bagel” occurs 400 times in a corpus of a million words
• What is the probability that a random word from some other text will be “bagel”?
• MLE estimate is 400/1,000,000 = .0004
• This may be a bad estimate for some other corpus
• But it is the estimate that makes it most likely that “bagel” will occur 400 times in a million word corpus.

Dan Jurafsky

Berkeley Restaurant Corpus: Laplace smoothed bigram counts

Dan Jurafsky

Laplace-smoothed bigrams

Dan Jurafsky

Reconstituted counts

Dan Jurafsky

Compare with raw bigram counts

Dan Jurafsky

Add-1 estimation is a blunt instrument

• So add-1 isn’t used for N-grams:
• We’ll see better methods
• But add-1 is used to smooth other NLP models
• For text classification
• In domains where the number of zeros isn’t so huge.

Dan Jurafsky

�Language Modeling

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Smoothing: Add-one (Laplace) smoothing

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Dan Jurafsky

�Language Modeling

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Interpolation, Backoff, and Web-Scale LMs

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Dan Jurafsky

Backoff and Interpolation

• Sometimes it helps to use less context
• Condition on less context for contexts you haven’t learned much about
• Backoff:
• use trigram if you have good evidence,
• otherwise bigram, otherwise unigram
• Interpolation:
• mix unigram, bigram, trigram

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• Interpolation works better

Dan Jurafsky

Linear Interpolation

• Simple interpolation

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• Lambdas conditional on context:

Dan Jurafsky

How to set the lambdas?

• Use a held-out corpus

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• Choose λs to maximize the probability of held-out data:
• Fix the N-gram probabilities (on the training data)
• Then search for λs that give largest probability to held-out set:

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Training Data

Held-Out Data

Test

Data

Dan Jurafsky

Unknown words: Open versus closed vocabulary tasks

• If we know all the words in advanced
• Vocabulary V is fixed
• Closed vocabulary task
• Often we don’t know this
• Out Of Vocabulary = OOV words
• Open vocabulary task
• Instead: create an unknown word token <UNK>
• Training of <UNK> probabilities
• Create a fixed lexicon L of size V
• At text normalization phase, any training word not in L changed to <UNK>
• Now we train its probabilities like a normal word
• At decoding time
• If text input: Use UNK probabilities for any word not in training

Dan Jurafsky

Huge web-scale n-grams

• How to deal with, e.g., Google N-gram corpus
• Pruning
• Only store N-grams with count > threshold.
• Remove singletons of higher-order n-grams
• Entropy-based pruning
• Efficiency
• Efficient data structures like tries
• Bloom filters: approximate language models
• Store words as indexes, not strings
• Use Huffman coding to fit large numbers of words into two bytes
• Quantize probabilities (4-8 bits instead of 8-byte float)

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Dan Jurafsky

Smoothing for Web-scale N-grams

• “Stupid backoff” (Brants et al. 2007)
• No discounting, just use relative frequencies

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Dan Jurafsky

N-gram Smoothing Summary

• Add-1 smoothing:
• OK for text categorization, not for language modeling
• The most commonly used method:
• Extended Interpolated Kneser-Ney
• For very large N-grams like the Web:
• Stupid backoff

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Dan Jurafsky

Advanced Language Modeling

• Discriminative models:
• choose n-gram weights to improve a task, not to fit the training set
• Parsing-based models
• Caching Models
• Recently used words are more likely to appear

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• These perform very poorly for speech recognition (why?)

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Dan Jurafsky

�Language Modeling

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Interpolation, Backoff, and Web-Scale LMs

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Dan Jurafsky