CMSC 723 / LING 645: Intro to Computational Linguistics

September 22, 2004: Dorr

Porter Stemmer,Intro to Probabilistic NLP and N-grams (chap 6.1-6.3)

Prof. Bonnie J. DorrDr. Christof Monz

TA: Adam Lee

Computational Morphology (continued)

The Rules and the Lexicon– General versus Specific– Regular versus Irregular– Accuracy, speed, space– The Morphology of a language

Approaches– Lexicon only– Lexicon and Rules

• Finite-state Automata• Finite-state Transducers

– Rules only

Lexicon-Free Morphology:Porter Stemmer

Lexicon-Free FST ApproachBy Martin Porter (1980)

http://www.tartarus.org/%7Emartin/PorterStemmer/

Cascade of substitutions given specific conditionsGENERALIZATIONS

GENERALIZATION

GENERALIZE

GENERAL

GENER

Porter Stemmer

Definitions– C = string of one or more consonants, where a consonant is anything other

than A E I O U or (Y preceded by C)– V = string of one or more vowels– M = Measure, roughly with number of syllables– Words = (C)*(V*C*)M(V)*

• M=0 TR, EE, TREE, Y, BY• M=1 TROUBLE, OATS, TREES, IVY• M=2 TROUBLES, PRIVATE, OATEN, ORRERY

Conditions– *S - stem ends with S– *v* - stem contains a V– *d - stem ends with double C, e.g., -TT, -SS – *o - stem ends CVC, where second C is not W, X or Y, e.g., -WIL, HOP

Porter Stemmer

Step 1: Plural Nouns and Third Person Singular VerbsSSES SS caresses caress

IES I ponies poni

ties ti

SS SS caress caress

S cats cat

*<S> = ends with <S>

*v* = contains a V

*d = ends with double C

*o = ends with CVC second C is not W, X or Y

Step 2a: Verbal Past Tense and Progressive Forms(M>0) EED EE feed feed, agreed agree

i (*v*) ED plastered plaster, bled bled

ii (*v*) ING motoring motor, sing sing

Step 2b: If 2a.i or 2a.ii is successful, Cleanup AT ATE conflat(ed) conflate

BL BLE troubl(ed) trouble

IZ IZE siz(ed) size

(*d and not (*L or *S or *Z)) hopp(ing) hop, tann(ed) tan

single letter hiss(ing) hiss, fizz(ed) fizz

(M=1 and *o) E fail(ing) fail, fil(ing) file

Porter Stemmer

Step 3: Y I

(*v*) Y I happy happi

sky sky

*<S> = ends with <S>

*v* = contains a V

*d = ends with double C

*o = ends with CVC second C is not W, X or Y

Porter Stemmer

Step 4: Derivational Morphology I: Multiple Suffixes (m>0) ATIONAL -> ATE relational -> relate (m>0) TIONAL -> TION conditional -> condition rational -> rational (m>0) ENCI -> ENCE valenci -> valence (m>0) ANCI -> ANCE hesitanci -> hesitance (m>0) IZER -> IZE digitizer -> digitize (m>0) ABLI -> ABLE conformabli -> conformable (m>0) ALLI -> AL radicalli -> radical (m>0) ENTLI -> ENT differentli -> different (m>0) ELI -> E vileli - > vile (m>0) OUSLI -> OUS analogousli -> analogous (m>0) IZATION -> IZE vietnamization -> vietnamize (m>0) ATION -> ATE predication -> predicate (m>0) ATOR -> ATE operator -> operate (m>0) ALISM -> AL feudalism -> feudal (m>0) IVENESS -> IVE decisiveness -> decisive (m>0) FULNESS -> FUL hopefulness -> hopeful (m>0) OUSNESS -> OUS callousness -> callous (m>0) ALITI -> AL formaliti -> formal (m>0) IVITI -> IVE sensitiviti -> sensitive (m>0) BILITI -> BLE sensibiliti -> sensible

Porter Stemmer

Step 5: Derivational Morphology II: More Multiple Suffixes (m>0) ICATE -> IC triplicate -> triplic

(m>0) ATIVE -> formative -> form

(m>0) ALIZE -> AL formalize -> formal

(m>0) ICITI -> IC electriciti -> electric

(m>0) ICAL -> IC electrical -> electric

(m>0) FUL -> hopeful -> hope

(m>0) NESS -> goodness -> good

Porter Stemmer

Step 6: Derivational Morphology III: Single Suffixes (m>1) AL -> revival -> reviv (m>1) ANCE -> allowance -> allow (m>1) ENCE -> inference -> infer (m>1) ER -> airliner -> airlin (m>1) IC -> gyroscopic -> gyroscop (m>1) ABLE -> adjustable -> adjust (m>1) IBLE -> defensible -> defens (m>1) ANT -> irritant -> irrit (m>1) EMENT -> replacement -> replac (m>1) MENT -> adjustment -> adjust (m>1) ENT -> dependent -> depend (m>1 and (*S or *T)) ION -> adoption -> adopt (m>1) OU -> homologou -> homolog (m>1) ISM -> communism -> commun (m>1) ATE -> activate -> activ (m>1) ITI -> angulariti -> angular (m>1) OUS -> homologous -> homolog (m>1) IVE -> effective -> effect (m>1) IZE -> bowdlerize -> bowdler

*<S> = ends with <S>

*v* = contains a V

*d = ends with double C

*o = ends with CVC second C is not W, X or Y

Porter Stemmer

Step 7a: Cleanup (m>1) E probate probat

rate rate

(m=1 and not *o) E cease ceas

Step 7b: More Cleanup

(m > 1 and *d and *L) controll control

single letter roll roll

*<S> = ends with <S>

*v* = contains a V

*d = ends with double C

*o = ends with CVC second C is not W, X or Y

Porter Stemmer

Errors of Omission– European Europe

– analysis analyzes

– matrices matrix

– noise noisy

– explain explanation

Errors of Commission– organization organ

– doing doe

– generalization generic

– numerical numerous

– university universe

From Krovetz ‘93

Why (not) Statistics for NLP?

Pro– Disambiguation– Error Tolerant– Learnable

Con– Not always appropriate– Difficult to debug

Weighted Automata/Transducers

Speech recognition: storing a pronunciation lexicon Augmentation of FSA: Each arc is associated with a

probability

Pronunciation network for “about”

Noisy Channel

Probability Definitions

Experiment (trial)– Repeatable procedure with well-defined possible

outcomesSample space

– Complete set of outcomesEvent

– Any subset of outcomes from sample spaceRandom Variable

– Uncertain outcome in a trial

More Definitions

Probability– How likely is it to get a particular outcome?– Rate of getting that outcome in all trials

Probability of drawing a spade from 52 well-shuffled playing cards:

Distribution: Probabilities associated with each outcome a random variable can take– Each outcome has probability between 0 and 1– The sum of all outcome probabilities is 1.

Conditional Probability

What is P(A|B)?First, what is P(A)?

– P(“It is raining”) = .06

Now what about P(A|B)?– P(“It is raining” | “It was clear 10 minutes ago”) = .004

)(

),()|(

BP

BAPBAP

A BA,B Note: P(A,B)=P(A|B) · P(B)Also: P(A,B) = P(B,A)

Independence

What is P(A,B) if A and B are independent?

P(A,B)=P(A) · P(B) iff A,B independent.– P(heads,tails) = P(heads) · P(tails) = .5 · .5 = .25– P(doctor,blue-eyes)

= P(doctor) · P(blue-eyes) = .01 · .2 = .002

What if A,B independent?– P(A|B)=P(A) iff A,B independent– Also: P(B|A)=P(B) iff A,B independent

Bayes Theorem

)(

)()|()|(

AP

BPBAPABP

• Swap the order of dependence

• Sometimes easier to estimate one kind of dependence than the other

What does this have to do with the Noisy Channel Model?

P(H)

(H) (O)

P(O|H) Best H

)(

)()|(

OP

HPHOPargmaxH

=Best H = argmax P(H|O)H

likelihood prior

Noisy Channel Applied to Word Recognition

argmaxw P(w|O) = argmaxw P(O|w) P(w)

Simplifying assumptions– pronunciation string correct– word boundaries known

Problem:– Given [n iy], what is correct dictionary word?

What do we need?[ni]: knee, neat, need, new

What is the most likely word given [ni]?

Now compute likelihood P([ni]|w), then multiplyWord P(O|w) P(w) P(O|w)P(w)

new .36 .001 .00036

neat .52 .00013 .000068

need .11 .00056 .000062

knee 1.00 .000024 .000024

Word freq(w) P(w)

new 2625 .001

neat 338 .00013

need 1417 .00056

knee 61 .000024

Compute prior P(w)

Why N-grams?

Word P(O|w) P(w) P(O|w)P(w)

new .36 .001 .00036

neat .52 .00013 .000068

need .11 .00056 .000062

knee 1.00 .000024 .000024

P([ni]|new)P(new)

P([ni]|neat)P(neat)

P([ni]|need)P(need)

P([ni]|knee)P(knee)

Unigram approach: ignores contextNeed to factor in context (n-gram)

- Use P(need|I) instead of just P(need)- Note: P(new|I) < P(need|I)

Compute likelihood P([ni]|w), then multiply

Next Word Prediction[borrowed from J. Hirschberg]

From a NY Times story...– 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.

Next Word Prediction (cont)

Human Word Prediction

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.

Compute:– probability of a sequence– likelihood of words co-occurring

Why would we want to do this?

Rank the likelihood of sequences containing various alternative alternative hypotheses

Assess the likelihood of a hypothesis

Why is this useful?

Speech recognitionHandwriting recognitionSpelling correctionMachine translation systemsOptical character recognizers

Handwriting Recognition

Assume a note is given to a bank teller, which the teller reads as I have a gub. (cf. Woody Allen)

NLP to the rescue ….– gub is not a word– gun, gum, Gus, and gull are words, but gun

has a higher probability in the context of a bank

Real Word Spelling Errors

They are leaving in about fifteen minuets to go to her house.

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.Can they lave him my messages? I need to notified the bank of….He is trying to fine out.

For Spell Checkers

Collect list of commonly substituted words– piece/peace, whether/weather, their/there ...

Example:“On Tuesday, the whether …’’“On Tuesday, the weather …”

Language Model

Definition: Language model is a model that enables one to compute the probability, or likelihood, of a sentence S, P(S).

Let’s look at different ways of computing P(S) in the context of Word Prediction

Word Prediction: Simple vs. Smart Simple:

Every word follows every other word w/ equal probability (0-gram)– Assume |V| is the size of the vocabulary– Likelihood of sentence S of length n is = 1/|V| × 1/|V| … × 1/|V|

– If English has 100,000 words, probability of each next word is 1/100000 = .00001

Smarter:Probability of each next word is related to word frequency (unigram)

– Likelihood of sentence S = P(w1) × P(w2) × … × P(wn)

– Assumes probability of each word is independent of probabilities of other words.

Even smarter: Look at probability given previous words (N-gram)

– Likelihood of sentence S = P(w1) × P(w2|w1) × … × P(wn|wn-1)

– Assumes probability of each word is dependent on probabilities of other words.

n times

Chain Rule

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) Examples:

– P(the dog) = P(the) P(dog | the)– P(the dog bites) = P(the) P(dog | the) P(bites| the dog)

Relative Frequencies and Conditional Probabilities

Relative word frequencies are better than equal probabilities for all words– In a corpus with 10K word types, each word would

have P(w) = 1/10K– Does not match our intuitions that different words are

more likely to occur (e.g. the)Conditional probability more useful than

individual relative word frequencies – Dog may be relatively rare in a corpus– But if we see barking, P(dog|barking) may be very large

For a Word String

In general, the probability of a complete string of words w1…wn is:

P(w )

= P(w1)P(w2|w1)P(w3|w1..w2)…P(wn|w1…wn-1)

= )11|

1( wk

n

kwkP

1n

But this approach to determining the probability of a word sequence is not very helpful in general….

Markov Assumption

How do we compute P(wn|w1n-1)?

Trick: Instead of P(rabbit|I saw a), we use P(rabbit|a).– This lets us collect statistics in practice

– A bigram model: P(the barking dog) = P(the|<start>)P(barking|the)P(dog|barking)

Markov models are the class of probabilistic models that assume that we can predict the probability of some future unit without looking too far into the past– Specifically, for N=2 (bigram): P(w1) ≈ Π P(wk|wk-1)

n n

k=1

Order of a Markov model: length of prior context– bigram is first order, trigram is second order, …

Counting Words in Corpora

What is a word? – e.g., are cat and cats the same word?– September and Sept?– zero and oh?– Is seventy-two one word or two? AT&T?– Punctuation?

How many words are there in English?Where do we find the things to count?

Corpora

Corpora are (generally online) collections of text and speech

Examples:– Brown Corpus (1M words)

– Wall Street Journal and AP News corpora

– ATIS, Broadcast News (speech)

– TDT (text and speech)

– Switchboard, Call Home (speech)

– TRAINS, FM Radio (speech)

Training and Testing

Probabilities come from a training corpus, which is used to design the model.– overly narrow corpus: probabilities don't generalize– overly general corpus: probabilities don't reflect task

or domainA separate test corpus is used to evaluate the

model, typically using standard metrics– held out test set– cross validation– evaluation differences should be statistically significant

Terminology

Sentence: unit of written languageUtterance: unit of spoken languageWord Form: the inflected form that appears in the

corpusLemma: lexical forms having the same stem, part

of speech, and word senseTypes (V): number of distinct words that might

appear in a corpus (vocabulary size) Tokens (N): total number of words in a corpusTypes seen so far (T): number of distinct words

seen so far in corpus (smaller than V and N)

Simple N-Grams

An N-gram model uses the previous N-1 words to predict the next one:P(wn | wn-N+1 wn-N+2… wn-1 )

– unigrams: P(dog)– bigrams: P(dog | big)– trigrams: P(dog | the big)– quadrigrams: P(dog | chasing the big)

Using N-Grams

Recall that– N-gram: P(wn|w1 ) ≈ P(wn|wn-N+1)

– Bigram: P(w1) ≈ Π P(wk|wk-1)n

n

k=1

n-1 n-1

For a bigram grammar– P(sentence) can be approximated by multiplying all the

bigram probabilities in the sequence

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)

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

Additional BERP Grammar

<start> I .25 Want some .04

<start> I’d .06 Want Thai .01

<start> Tell .04 To eat .26

<start> I’m .02 To have .14

I want .32 To spend .09

I would .29 To be .02

I don’t .08 British food .60

I have .04 British restaurant .15

Want to .65 British cuisine .01

Want a .05 British lunch .01

Computing Sentence Probability

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 = .00015Probabilities seem to capture “syntactic” facts,

“world knowledge”– eat is often followed by a NP– British food is not too popular

N-gram models can be trained by counting and normalization

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

BERP Bigram Probabilities: Use Unigram Count

Normalization: divide bigram count by unigram count of first word.

I Want To Eat Chinese Food Lunch

3437 1215 3256 938 213 1506 459

Computing the probability of I I– P(I|I) = C(I,I)/C(I) = 8 / 3437 = .0023

A bigram grammar is an NxN matrix of probabilities, where N is the vocabulary size

Learning a Bigram Grammar

The formula P(wn|wn-1) = C(wn,wn-1)/C(wn-1) is used for bigram “parameter estimation”

Relative FrequencyMaximum Likelihood Estimation (MLE):

Parameter set maximizes likelihood of training set T given model M — P(T|M).

What do we learn about the language?

What about...– P(I | I) = .0023

– P(I | want) = .0025

– P(I | food) = .013

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

Readings for next time

J&M Chapter 5, 7.1-7.3