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Lecture 1, 7/21/2005 Natural Language Processing 1
CS60057Speech &Natural Language
Processing
Autumn 2007
Lecture 5
2 August 2007
Lecture 1, 7/21/2005 Natural Language Processing 2
WORDSThe Building Blocks of Language
Lecture 1, 7/21/2005 Natural Language Processing 3
Language can be divided up into pieces of varying sizes, ranging from morphemes to paragraphs.
Words -- the most fundamental level for NLP.
Lecture 1, 7/21/2005 Natural Language Processing 4
Tokens, Types and Texts
This process of segmenting a string of characters into words is known as tokenization.
>>> sentence = "This is the time -- and this is the record of the time." >>> sentence = "This is the time -- and this is the record of the time." >>> words = sentence.split() >>> words = sentence.split() >>> len(words) >>> len(words) 13
Compile a list of the unique vocabulary items in a string by using set() to eliminate duplicates
>>> len(set(words)) >>> len(set(words)) 10 10
A word token is an individual occurrence of a word in a concrete context.A word type is what we're talking about when we say that the three occurrences
of the in sentence are "the same word."
Lecture 1, 7/21/2005 Natural Language Processing 5
>>> set(words) set(['and', 'this', 'record', 'This', 'of', 'is', '--', 'time.', 'time', 'the']
Extracting text from files >>> f = open('corpus.txt', 'rU') >>> f.read() 'Hello World!\nThis is a test file.\n'
We can also read a file one line at a time using the for loop construct: >>> f = open('corpus.txt', 'rU') >>> for line in f: ... print line[:-1] Hello world! This is a test file.
Here we use the slice [:-1] to remove the newline character at the end of the input line.
Lecture 1, 7/21/2005 Natural Language Processing 6
Extracting text from the Web
>>> from urllib import urlopen >>> page = urlopen("http://news.bbc.co.uk/").read() >>> print page[:60] <!doctype html public "-//W3C//DTD HTML 4.0 Transitional//EN"
Web pages are usually in HTML format. To extract the text, we need to strip out the HTML markup, i.e. remove all material enclosed in angle brackets. Let's digress briefly to consider how to carry out this task using regular expressions. Our first attempt might look as follows:
>>> line = '<title>BBC NEWS | News Front Page</title>‘>>> new = re.sub(r'<.*>', '', line) >>> new ‘ '
Lecture 1, 7/21/2005 Natural Language Processing 7
What has happened here? 1. The wildcard '.' matches any character other than '\n', so it will match '>'
and '<'. 2. The '*' operator is "greedy", it matches as many characters as it can. In the
above example, '.*' will return not the shortest match, namely 'title', but the longest match, 'title>BBC NEWS | News Front Page</title'. To get the shortest match we have to use the '*?' operator. We will also normalise whitespace, replacing any sequence of one or more spaces, tabs or newlines (these are all matched by '\s+') with a single space character:
>>> page = re.sub('<.*?>', '', page) >>> page = re.sub('\s+', ' ', page) >>> print page[:60] BBC NEWS | News Front Page News Sport Weather World Service
Lecture 1, 7/21/2005 Natural Language Processing 8
Extracting text from NLTK Corpora
NLTK is distributed with several corpora and corpus samples and many are supported by the corpus package.
>>> corpus.gutenberg.items ['austen-emma', 'austen-persuasion', 'austen-sense', 'bible-kjv', 'blake-
poems', 'blake-songs', 'chesterton-ball', 'chesterton-brown', 'chesterton-thursday', 'milton-paradise', 'shakespeare-caesar', 'shakespeare-hamlet', 'shakespeare-macbeth', 'whitman-leaves']
Next we iterate over the text content to find the number of word tokens: >>> count = 0 >>> for word in corpus.gutenberg.read('whitman-leaves'): ... count += 1 >>> print count 154873
Lecture 1, 7/21/2005 Natural Language Processing 9
Brown Corpus
The Brown Corpus was the first million-word, part-of-speech tagged electronic corpus of English, created in 1961 at Brown University. Each of the sections a through r represents a different genre.
>>> corpus.brown.items ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'j', 'k', 'l', 'm', 'n', 'p', 'r'] >>> corpus.brown.documents['a'] 'press: reportage' We can extract individual sentences (as lists of words) from the corpus
using the read() function. Here we will specify section a, and indicate that only words (and not part-of-speech tags) should be produced.
>>> a = corpus.brown.tokenized('a') >>> a[0] ['The', 'Fulton', 'County', 'Grand', 'Jury', 'said', 'Friday', 'an', 'investigation',
'of', "Atlanta's", 'recent', 'primary', 'election', 'produced', '``', 'no', 'evidence', "''", 'that', 'any', 'irregularities', 'took', 'place', '.']
Lecture 1, 7/21/2005 Natural Language Processing 10
Lecture 1, 7/21/2005 Natural Language Processing 11
Corpus Linguistics
1. Text-corpora: Brown corpus. One million words, tagged, representative of American English.
2. Text-corpora: Project Gutenberg. 17,000 uncopyrighted literary texts (Tom Sawyer, etc.)
3. Text-corpora: OMIM: Comprehensive list of medical conditions. 4. Word frequencies. 5. Zipf's First Law.
Lecture 1, 7/21/2005 Natural Language Processing 12
What’s a word?
I have a can opener; but I can’t open these cans. how many words?
Word form inflected form as it appears in the text can and cans ... different word forms
Lemma a set of lexical forms having the same stem, same POS and same meaning can and cans … same lemma
Word token: an occurrence of a word I have a can opener; but I can’t open these cans. 11 word tokens (not counting
punctuation)
Word type: a different realization of a word I have a can opener; but I can’t open these cans. 10 word types (not counting
punctuation)
Lecture 1, 7/21/2005 Natural Language Processing 13
Another example
Mark Twain’s Tom Sawyer 71,370 word tokens 8,018 word types tokens/type ratio = 8.9 (indication of text complexity)
Complete Shakespeare work 884,647 word tokens 29,066 word types tokens/type ratio = 30.4
Lecture 1, 7/21/2005 Natural Language Processing 14
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?
Lecture 1, 7/21/2005 Natural Language Processing 15
Common words in Tom Sawyer
but words in NL have an uneven distribution…
Lecture 1, 7/21/2005 Natural Language Processing 16
Text properties (formalized)Sample word frequency data
Lecture 1, 7/21/2005 Natural Language Processing 17
Frequency of frequencies most words are rare
3993 (50%) word types appear only once they are called happax legomena (read
only once)
but common words are very common 100 words account for 51% of all tokens
(of all text)
Lecture 1, 7/21/2005 Natural Language Processing 18
Zipf’s Law
1. Count the frequency of each word type in a large corpus2. List the word types in order of their frequency Let:
f = frequency of a word type r = its rank in the list
Zipf’s Law says: f 1/r In other words:
there exists a constant k such that: f × r = k The 50th most common word should occur with 3 times the
frequency of the 150th most common word.
Lecture 1, 7/21/2005 Natural Language Processing 19
Zipf’s Law
If probability of word of rank r is pr and N is the total number of word occurrences:
1.0 const. indp. corpusfor ArA
Nfpr
Lecture 1, 7/21/2005 Natural Language Processing 20
Zipf curve
Lecture 1, 7/21/2005 Natural Language Processing 21
Predicting Occurrence Frequencies By Zipf, a word appearing n times has rank rn=AN/n If several words may occur n times, assume rank rn applies to the last of these. Therefore, rn words occur n or more times and rn+1 words occur n+1 or more times. So, the number of words appearing exactly n times is:
)1(11
nn
ANnAN
nANrrI nnn
Fraction of words with frequency n is:
Fraction of words appearing only once is therefore ½.)1(
1
nnD
In
Lecture 1, 7/21/2005 Natural Language Processing 22
Explanations for Zipf’s Law
- Zipf’s explanation was his “principle of least effort.” Balance between speaker’s desire for a small vocabulary and hearer’s desire for a large one.
Lecture 1, 7/21/2005 Natural Language Processing 23
Zipf’s First Law
1. f 1/r∝ , f = word-frequency, r = word-frequency rank, m = number of meetings per word.
2. There exists a k such that f × r = k. 3. Alternatively, log f = log k - log r. 4. English literature, Johns Hopkins Autopsy Resource, German,
and Chinese. 5. Most famous of Zipf’s Laws.
Lecture 1, 7/21/2005 Natural Language Processing 24
Zipf’s Second Law
1. Meanings, m √f∝ 2. There exists a k such that k × f = m2. 3. Corollary: m 1/√r∝
Lecture 1, 7/21/2005 Natural Language Processing 25
Zipf’s Third Law
1. Frequency ∝ 1/wordlength: 2. There exists a k such that f × wordlength = k. 3. Many other minor laws stated.
Lecture 1, 7/21/2005 Natural Language Processing 26
Zipf’s Law Impact on Language Analysis
Good News: Stopwords will account for a large fraction of text so eliminating them greatly reduces size of vocabulary in a text
Bad News: For most words, gathering sufficient data for meaningful statistical analysis (e.g. for correlation analysis for query expansion) is difficult since they are extremely rare.
Lecture 1, 7/21/2005 Natural Language Processing 27
Vocabulary Growth
How does the size of the overall vocabulary (number of unique words) grow with the size of the corpus?
This determines how the size of the inverted index will scale with the size of the corpus.
Vocabulary not really upper-bounded due to proper names, typos, etc.
Lecture 1, 7/21/2005 Natural Language Processing 28
Heaps’ Law
If V is the size of the vocabulary and the n is the length of the corpus in words:
Typical constants: K 10100 0.40.6 (approx. square-root)
10 , constants with KKnV
Lecture 1, 7/21/2005 Natural Language Processing 29
Heaps’ Law Data
Lecture 1, 7/21/2005 Natural Language Processing 30
Word counts are interesting...
As an indication of a text’s style As an indication of a text’s author
But, because most words appear very infrequently, it is hard to predict much about the behavior of words
(if they do not occur often in a corpus) --> Zipf’s Law
Lecture 1, 7/21/2005 Natural Language Processing 31
Zipf’s Law on Tom Saywer
k ≈ 8000-9000 except for
The 3 most frequent wordsWords of frequency ≈ 100
Lecture 1, 7/21/2005 Natural Language Processing 32
Plot of Zipf’s LawOn chap. 1-3 of Tom Sawyer (≠ numbers from p. 25&26)
f×r = k
Zipf
0
50
100
150
200
250
300
350
0 500 1000 1500 2000
Rank
Freq
Lecture 1, 7/21/2005 Natural Language Processing 33
Plot of Zipf’s Law (con’t)On chap. 1-3 of Tom Sawyer f×r = k ==> log(f×r) = log(k) ==> log(f)+log(r) = log(k)
Zipf's Law
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8
log(rank)
log(
freq
)
Lecture 1, 7/21/2005 Natural Language Processing 34
Zipf’s Law, so what?
There are: A few very common words A medium number of medium frequency words A large number of infrequent words
Principle of Least effort: Tradeoff between speaker and hearer’s effort Speaker communicates with a small vocabulary of common words (less
effort) Hearer disambiguates messages through a large vocabulary of rare
words (less effort)
Significance of Zipf’s Law for us: For most words, our data about their use will be very sparse Only for a few words will we have a lot of examples
Lecture 1, 7/21/2005 Natural Language Processing 35
N-Grams and Corpus Linguistics
Lecture 1, 7/21/2005 Natural Language Processing 36
A bad language model
N-grams & Language Modeling
Lecture 1, 7/21/2005 Natural Language Processing 37
A bad language model
Lecture 1, 7/21/2005 Natural Language Processing 38
A bad language model
Herm
an is reprinted with perm
ission from LaughingStock Licensing Inc., O
ttawa C
anada. A
ll rights reserved.
Lecture 1, 7/21/2005 Natural Language Processing 39
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
Lecture 1, 7/21/2005 Natural Language Processing 40
What’s a language model for?
Speech recognition Handwriting recognition Spelling correction Optical character recognition Machine translation
(and anyone doing statistical modeling)
Lecture 1, 7/21/2005 Natural Language Processing 41
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
Lecture 1, 7/21/2005 Natural Language Processing 42
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.
Lecture 1, 7/21/2005 Natural Language Processing 43
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
Lecture 1, 7/21/2005 Natural Language Processing 44
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)
Lecture 1, 7/21/2005 Natural Language Processing 45
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 ASRTheatre owners say popcorn/unicorn sales have
doubled... Assess the likelihood/goodness of a sentence, e.g. for
text generation or machine translationThe doctor recommended a cat scan.El doctor recommendó una exploración del gato.
Lecture 1, 7/21/2005 Natural Language Processing 47
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
Lecture 1, 7/21/2005 Natural Language Processing 48
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
Lecture 1, 7/21/2005 Natural Language Processing 49
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
Lecture 1, 7/21/2005 Natural Language Processing 50
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
Lecture 1, 7/21/2005 Natural Language Processing 51
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 bigramsP(the,mythical,unicorn) = P(unicorn|mythical) P(mythical|
the) P(the|<start>)
)11|( nn wwP )1
1|(
nNnn wwP
n
kkkn wwPwP
111 )|()(
Lecture 1, 7/21/2005 Natural Language Processing 52
The n-gram ApproximationAssume 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”)
Lecture 1, 7/21/2005 Natural Language Processing 53
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”)
Lecture 1, 7/21/2005 Natural Language Processing 54
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.
Lecture 1, 7/21/2005 Natural Language Processing 55
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.
Lecture 1, 7/21/2005 Natural Language Processing 56
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
Lecture 1, 7/21/2005 Natural Language Processing 57
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?
Lecture 1, 7/21/2005 Natural Language Processing 58
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
Lecture 1, 7/21/2005 Natural Language Processing 59
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
Lecture 1, 7/21/2005 Natural Language Processing 60
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
Lecture 1, 7/21/2005 Natural Language Processing 61
Computing the Probability of a Word Sequence
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
Lecture 1, 7/21/2005 Natural Language Processing 62
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
Lecture 1, 7/21/2005 Natural Language Processing 63
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 bigramsP(the,mythical,unicorn) = P(unicorn|mythical) P(mythical|
the) P(the|<start>)
)11|( nn wwP )1
1|(
nNnn wwP
n
kkkn wwPwP
111 )|()(
Lecture 1, 7/21/2005 Natural Language Processing 64
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
Lecture 1, 7/21/2005 Natural Language Processing 65
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)
Lecture 1, 7/21/2005 Natural Language Processing 66
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
Lecture 1, 7/21/2005 Natural Language Processing 67
.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
Lecture 1, 7/21/2005 Natural Language Processing 68
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
Lecture 1, 7/21/2005 Natural Language Processing 69
BERP Bigram Counts
0100004Lunch
000017019Food
112000002Chinese
522190200Eat
12038601003To
686078603Want
00013010878I
lunchFoodChineseEatToWantI
Lecture 1, 7/21/2005 Natural Language Processing 70
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
Lecture 1, 7/21/2005 Natural Language Processing 71
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
Lecture 1, 7/21/2005 Natural Language Processing 72
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 ...
Lecture 1, 7/21/2005 Natural Language Processing 73
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.
Lecture 1, 7/21/2005 Natural Language Processing 74
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?
Lecture 1, 7/21/2005 Natural Language Processing 75
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
Lecture 1, 7/21/2005 Natural Language Processing 76
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
Lecture 1, 7/21/2005 Natural Language Processing 77
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?
Lecture 1, 7/21/2005 Natural Language Processing 78
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?
Lecture 1, 7/21/2005 Natural Language Processing 79
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?
Lecture 1, 7/21/2005 Natural Language Processing 80
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
VNNci
1
VNc
ipi
1*
Lecture 1, 7/21/2005 Natural Language Processing 81
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!
Lecture 1, 7/21/2005 Natural Language Processing 82
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
View training corpus as series of events, one for each token (N) and one for each new type (T)TN
T
Witten-Bell Discounting
Lecture 1, 7/21/2005 Natural Language Processing 83
We can divide the probability mass equally among unseen bigrams….or we can condition the probability of an unseen bigram on the first word of the bigram
Discount values for Witten-Bell are much more reasonable than Add-One
Lecture 1, 7/21/2005 Natural Language Processing 84
Re-estimate amount of probability mass for zero (or low count) ngrams by looking at ngrams with higher counts Estimate
E.g. N0’s adjusted count is a function of the count of ngrams that occur once, N1
Assumes: word bigrams follow a binomial distribution We know number of unseen bigrams (VxV-seen)
NcNccc 11*
Good-Turing Discounting
Lecture 1, 7/21/2005 Natural Language Processing 85
Backoff methods (e.g. Katz ‘87)
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
E.g An omnivorous unicorn
Lecture 1, 7/21/2005 Natural Language Processing 86
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