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Language Modeling

Page 2

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

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• 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.

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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

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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)

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N-grams

N-grams ‘N-words’ ?

It’s ‘N’ consecutive words that one can find in a given corpus or set of documents ?

‘N-gram’ model is a probabilistic model that computes probability of Nth word occurring after seeing ‘N-1’ words – also known as language model in speech recognition

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Some Interesting Facts

The most frequent 250 words takes account of approximately 50% of all tokens in any random text

‘the’ is usually the most frequent word

‘the’ occurs 69,971 times in 1 million word Brown corpus (7%)

The top 20 words in 1 year of Wall Street Journal is

• The, of, to, in, and, for, that, is, on, it, by, with, as, at, said, mister, from, its, are, he, million

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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

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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.

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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 ?

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Terminology

Sentence: unit of written language

Utterance: unit of spoken language

Types: number of distinct words in a corpus (vocabulary size)

Tokens: total number of words

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Corpora

Corpora are collections of text and speech

• Brown Corpus

• Wall Street Journal

• AP news

• DARPA/NIST text/speech corpora (Call Home, ATIS, switchboard, Broadcast News, TDT, Communicator)

• TRAINS, Radio News

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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

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Computing the Probability of a Word Sequence

Conditional Probability

• P(A1,A2) = P(A1) P(A2|A1)

The Chain Rule generalizes to multiple events

• P(A1, …,An) = P(A1) P(A2|A1) P(A3|A1,A2)…P(An|A1…An-1)

Compute the product of component conditional probabilities?• 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

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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

)11|( nn wwP )|( 1nn wwP

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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>)

)11|( nn wwP )1

1|(

nNnn wwP

n

kkkn wwPwP

111 )|()(

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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

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A Simple Example

• P(I want to eat Chinese food) = P(I | <start>) P(want | I) P(to | want) P(eat | to) P(Chinese | eat) P(food | Chinese)

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A Bigram Grammar Fragment from BERP

.001Eat British.03Eat today

.007Eat dessert.04Eat Indian

.01Eat tomorrow.04Eat a

.02Eat Mexican.04Eat at

.02Eat Chinese.05Eat dinner

.02Eat in.06Eat lunch

.03Eat breakfast.06Eat some

.03Eat Thai.16Eat on

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.01British lunch.05Want a

.01British cuisine.65Want to

.15British restaurant.04I have

.60British food.08I don’t

.02To be.29I would

.09To spend.32I want

.14To have.02<start> I’m

.26To eat.04<start> Tell

.01Want Thai.06<start> I’d

.04Want some.25<start> I

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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

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BERP Bigram Counts

0100004Lunch

000017019Food

112000002Chinese

522190200Eat

12038601003To

686078603Want

00013010878I

lunchFoodChineseEatToWantI

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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.

4591506213938325612153437

LunchFoodChineseEatToWantI

)()(

1

2,1

wfreqwwfreq

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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

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• 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 ...

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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.

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Trigrams

• 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?

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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

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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

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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?

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Smoothing Techniques

Every n-gram training matrix is sparse, even for very large corpora

Solution: estimate the likelihood of unseen n-grams

Problems: how do you adjust the rest of the corpus to accommodate these ‘phantom’ n-grams?

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Add-one SmoothingFor 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:VNNci

1

VNc

ipi

1*

VxyC

xyzCxyzP

)(

1)()|(

VxyC

xyzCxyzP

)(

)()|(Add delta smoothing

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Basic Idea: Use Count of things you have seen once to estimate the things you haven’t seen

• i.e. Estimate the probability of seeing something for the first time

View training corpus as series of events, one for each token (N) and one for each new type (T). Probability of seeing something new is

This probability mass will assigned to unseen cases

TNT

Witten-Bell Discounting

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Witten-Bell (cont…)A zero ngram is just an ngram you haven’t seen yet…but every ngram in the corpus was unseen once…so...

• How many times did we see an ngram for the first time? Once for each ngram type (T)

• Est. total probability of unseen bigrams as

• Each of Z unseen cases will be assigned an equal portion probability mass - T/N+T

• Seen cases will have probabilities discounted as

TNT

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Witten-Bell (cont…)

But for bigrams we can condition on the first word:

• Instead of trying to find what is the probability of finding a new bigram we can ask what is the probability of finding a new bigram that starts with the given word w

Then unseen portion will be assigned probability mass

And for seen cases we discount as follows:

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Distributing Among the Zeros

If a bigram “wx wi” has a zero count

)()(

)(

)(

1)|(

xx

x

xxi

wTwN

wT

wZwwP

Number of bigrams starting with wx that were not seen

Actual frequency of bigrams beginning with wx

Number of bigram types starting with wx

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Re-estimate amount of probability mass for zero (or low count) ngrams by looking at ngrams with higher counts• For any n-gram that occurs ‘c’ times assume it occurs c* times,

where Nc is the n number n-grams occurring c times

• Estimate a smoothed count

• E.g. N0’s adjusted count is a function of the count of ngrams that occur once, N1

• Probability mass assigned to unseen cases works out to be n1/N

• Counts for bigrams that never occurred (c0) will be just count of bigrams that occurred once by count of bigrams that never occurred. How do we know count of bigrams that never occurred?

Nc

Nccc 11*

Good-Turing Discounting

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Backoff methods (e.g. Katz ‘87)

If you don’t have a count for ‘n-gram’ backoff to weighted ‘n-1’ gram

Alphas needed to make a proper probability distribution (If we backoff when probability is zero, we are adding extra probability mass)

For e.g. a trigram model• Compute unigram, bigram and trigram probabilities

• In use:– Where trigram unavailable back off to bigram if available, o.w. unigram

probability

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Summary

N-gram models are approximation to the correct model given by chain rule

N-gram probabilities can be used to estimate the likelihood

• Of a word occurring in a context (N-1)

N-gram models suffer from sparse data

Smoothing techniques deal with problems of unseen words in corpus

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Statistical Techniques in NLP

Page 41

Example: Text Summarization

Let’s say we are doing text summarization using sentence extraction

Someone gave us a document where each sentence is scored between 1 to 20 and

Extracting top 10% scoring sentence can potentially be a summary

We have 100 such documents

Problem: Use a machine learning technique to build a model that predicts the score of sentences in a new document

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Example: Text Summarization

CORPUS100 documentsEach sentence scored (1 to 20)

MachineLearning

Corpus-BasedModel

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Regression for our Text Summarization Problem

For simplicity let’s assume all of documents are of equal length ‘N’

Our training dataset (100 documents)

• We have 100xN sentences in total

• 100xN scores, let’s call these scores y’s

• For each sentence we know it’s position in the document. Let’s call these sentence positions x’s so we get 100xN sentence positions

)},(),,)...(,(),,{( 100100100)1(100)1(2211 nxnxxnxn yxyxyxyxX

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Regression as Function Approximation

Using our training data ‘X’ we must predict a score for each sentence in a new document

We can do such prediction by finding a function y=f(x) that fits the training data well

• Need to find out what f(x) to use

• Need to compute how good the training data fits with the chosen f(x)

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Empirical Risk Minimization

Need to compute how good the data fits the chosen function f(x)

We can define a loss function L(y, f(x))

Find average loss:

Simple Loss Function:

N

iiiemp xfyL

NR

1

))(,(1

2))(())(,( iiii xfyxfyL

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Linear Regression

We can choose f(x) to be linear, polynomial, exponential or any other class of functions

If we use linear function we get linear regression, widely used modeling technique

01);( xxf

cmxy Where m is slope and c is intercept on y axis

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Text Summarization with Linear Regression

We had Nx100 (xi, yi) pairs where x was sentence position and y was score for the sentence

Let’s look at a sample plot

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9 10

Series1

Sentence position

Score(xi, yi)

(xi, f(xi))

Error (yi - f(xi))

The line we have found by minimizing average squared error is our model (summarizer)

Given any new ‘x’ (i.e. sentence position of a new document) we can predict ‘y’ – score that represents it’s significance to summary

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Naïve Bayes Classifier

Let’s assume instead of score between 1 to 20 for each sentence, all the sentence have been classified into two classes – IN SUMMARY (1) , NOT IN SUMMARY (0)

Now, given a document we want to predict the class (1) or (0) for each sentence in the document. All the sentence in class (1) should be included in the summary

This is a binary classification problem

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Intuitive Example of Naïve Bayes Classifier

• Let us assume the objects can be classified into RED or GREEN

• We have a corpus with objected manually labeled as RED or GREEN

• We need to figure out if the test object is RED or GREEN• Number of green objects is twice that of red. So, it is

reasonable to assume that a new object (not observed yet) is twice likely to be green than red (prior probability)

All figures in the given example are from electronic textbook StatSoft

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Example (RED or GREEN classification)

There are total of 60 objects with 40 GREEN and 20 RED

Prior probability of GREEN # of GREEN objectsTotal # of objects

Prior probability of RED # of RED objectsTotal # of objects

Prior probability of GREEN 4060

Prior probability of RED 2060

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Example (RED or GREEN classification)

Looking at the picture it is reasonable to assume that likelihood of object being RED or GREEN can be computed by number of RED or GREEN objects in the vicinity

Need to figure out if thetest object is GREEN or RED

Likelihood of X being GREEN # of GREEN objects in vicinity of XTotal # of GREEN

Likelihood of X being RED # of RED objects in vicinity of XTotal # of RED

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Example (RED or GREEN classification)

Looking at the picture it is reasonable to assume that likelihood of object being RED or GREEN can be computed by number of RED or GREEN objects in the vicinity

Need to figure out if thetest object is GREEN or RED

Likelihood of X being GREEN # of GREEN objects in vicinity of XTotal # of GREEN

Likelihood of X being RED # of RED objects in vicinity of XTotal # of RED

1/40

3/20

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Example (RED or GREEN classification)

Posterior probability of X being GREEN = Prior probability of GREEN x Likelihood of X given GREEN

= 40/60 x 1/40 = 1/60

Posterior probability of X being RED = Prior probability of RED x Likelihood of X

given RED = 20/60 x 3/40 = 1/40

Hence, we classify our new object X as RED using ourBayesian classifier model.

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Document Vectors

Document Vectors

• Documents can be represented in different types of vectors: binary vector, multinomial vector, feature vector

Binary Vector: For each dimension, 1 if the word type is in the document and 0 otherwise

Multinomial Vector: For each dimension, count # of times word type appears in the document

Feature Vector: Extract various features from the document and represent them in a vector. Dimension equals the number of features

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Example of a multinomial document vector

Screening of the critically acclaimed film NumaFung Reserved tickets can be picked up on the day of the show at the box office at Arledge Cinema. Tickets will not be reserved if not paid for in advance.

4 THE 2 TICKETS 2 RESERVED 2 OF 2 NOT 2 BE 2 AT 1 WILL 1 UP 1 SHOW 1 SCREENING 1 PICKED 1 PAID 1 ON 1 OFFICE 1 NUMAFUNG 1 IN 1 IF 1 FOR 1 FILM 1 DAY 1 CRITICALLY 1 CINEMA 1 CAN 1 BOX 1 ARLEDGE 1 ADVANCE 1 ACCLAIMED

4 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Example of a multinomial document vector

4 THE 2 SEATS 2 RESERVED 2 OF 2 NOT 2 BE 2 AT 1 WILL 1 UP 1 SHOW 1 SHOWING1 PICKED 1 PAID 1 ON 1 OFFICE 1 VOLCANO 1 IN 1 IF 1 FOR 1 FILM 1 DAY 1 CRITICALLY 1 CINEMA 1 CAN

4 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

But if you want to compare previous vector to this new vector we cannot do any computation like cosine measure with this vector. Why?

->Different dimensions

Hence, if you have multiple documents you need to first find set of all words (dimensions) in the set of documents. Values for all the words that does not appear in the given document is 0

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Feature Vectors

Instead of just using words to represent documents, we can also extract features and use them to represent the document

For example, let’s classify History and Physics documents

We are given ‘N’ documents which are labeled as ‘HISTORY’ or ‘PHYSICS’

We can extract features like document length (LN), number of nouns (NN), number of verbs (VB), number of person names (PN), number of place (CN) names, number of organization names (ON), number of sentences (NS), number of pronouns (PNN)

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Feature Vectors

Extracting such features you get a feature vector of length ‘K’ where ‘K’ is the number of dimensions (features) for each document

LengthNoun CountVerb Count# Person

Name# Place Name

# Orgzn Name

.

.

.

7804534 76723

.

.

.

42012560 734

.

.

.

…

9403638 65521

.

.

.

HIST PHYSICS HIST

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Feature Vectors

LengthNoun CountVerb Count# Person

Name# Place Name

# Orgzn Name

.

.

.

7804534 76723

.

.

.

42012560 734

.

.

.

…

9403638 65521

.

.

.

HIST PHYSICS HIST After such feature vector representation we can use various learning algorithms including Naïve Bayes, Decision Trees, to classify the new document using the training document set

9403638 65521

.

.

.

HIST