Lecture 1, 7/21/2005Natural Language Processing1 CS60057 Speech &Natural Language Processing Autumn 2007 Lecture 5 2 August 2007
Lecture 1, 7/21/2005Natural Language Processing2 WORDS The Building Blocks of Language
Lecture 1, 7/21/2005Natural Language Processing3 Language can be divided up into pieces of varying sizes, ranging from morphemes to paragraphs. Words -- the most fundamental level for NLP.
Lecture 1, 7/21/2005Natural Language Processing4 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." >>> words = sentence.split() >>> len(words) 13 Compile a list of the unique vocabulary items in a string by using set() to eliminate duplicates >>> len(set(words)) 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, 7/21/2005Natural Language Processing5 >>> 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, 7/21/2005Natural Language Processing6 Extracting text from the Web >>> from urllib import urlopen >>> page = urlopen(" >>> 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 = ' BBC NEWS | News Front Page ‘ >>> new = re.sub(r' ', '', line) >>> new ‘ '
Lecture 1, 7/21/2005Natural Language Processing7 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, 7/21/2005Natural Language Processing8 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
Lecture 1, 7/21/2005Natural Language Processing9 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, 7/21/2005Natural Language Processing10
Lecture 1, 7/21/2005Natural Language Processing11 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, 7/21/2005Natural Language Processing12 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, 7/21/2005Natural Language Processing13 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, 7/21/2005Natural Language Processing14 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, 7/21/2005Natural Language Processing15 Common words in Tom Sawyer but words in NL have an uneven distribution…
Lecture 1, 7/21/2005Natural Language Processing16 Text properties (formalized) Sample word frequency data
Lecture 1, 7/21/2005Natural Language Processing17 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, 7/21/2005Natural Language Processing18 Zipf’s Law 1. Count the frequency of each word type in a large corpus 2. 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 50 th most common word should occur with 3 times the frequency of the 150 th most common word.
Lecture 1, 7/21/2005Natural Language Processing19 Zipf’s Law If probability of word of rank r is p r and N is the total number of word occurrences:
Lecture 1, 7/21/2005Natural Language Processing20 Zipf curve
Lecture 1, 7/21/2005Natural Language Processing21 Predicting Occurrence Frequencies By Zipf, a word appearing n times has rank r n =AN/n If several words may occur n times, assume rank r n applies to the last of these. Therefore, r n words occur n or more times and r n+1 words occur n+1 or more times. So, the number of words appearing exactly n times is: Fraction of words with frequency n is: Fraction of words appearing only once is therefore ½.
Lecture 1, 7/21/2005Natural Language Processing22 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, 7/21/2005Natural Language Processing23 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, 7/21/2005Natural Language Processing24 Zipf’s Second Law 1. Meanings, m ∝ √f 2. There exists a k such that k × f = m 2. 3. Corollary: m ∝ 1/√r
Lecture 1, 7/21/2005Natural Language Processing25 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, 7/21/2005Natural Language Processing26 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, 7/21/2005Natural Language Processing27 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, 7/21/2005Natural Language Processing28 Heaps’ Law If V is the size of the vocabulary and the n is the length of the corpus in words: Typical constants: K 10 100 0.4 0.6 (approx. square-root )
Lecture 1, 7/21/2005Natural Language Processing29 Heaps’ Law Data
Lecture 1, 7/21/2005Natural Language Processing30 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, 7/21/2005Natural Language Processing31 Zipf’s Law on Tom Saywer k ≈ except for The 3 most frequent words Words of frequency ≈ 100
Lecture 1, 7/21/2005Natural Language Processing32 Plot of Zipf’s Law On chap. 1-3 of Tom Sawyer (≠ numbers from p. 25&26) f×r = k
Lecture 1, 7/21/2005Natural Language Processing33 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)
Lecture 1, 7/21/2005Natural Language Processing34 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, 7/21/2005Natural Language Processing35 N-Grams and Corpus Linguistics
Lecture 1, 7/21/2005Natural Language Processing36 A bad language model N-grams & Language Modeling
Lecture 1, 7/21/2005Natural Language Processing37 A bad language model
Lecture 1, 7/21/2005Natural Language Processing38 A bad language model Herman is reprinted with permission from LaughingStock Licensing Inc., Ottawa Canada. Allrights reserved.
Lecture 1, 7/21/2005Natural Language Processing39 What’s a Language Model A Language model is a probability distribution over word sequences P(“And nothing but the truth”) P(“And nuts sing on the roof”) 0
Lecture 1, 7/21/2005Natural Language Processing40 What’s a language model for? Speech recognition Handwriting recognition Spelling correction Optical character recognition Machine translation (and anyone doing statistical modeling)
Lecture 1, 7/21/2005Natural Language Processing41 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, 7/21/2005Natural Language Processing42 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, 7/21/2005Natural Language Processing43 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, 7/21/2005Natural Language Processing44 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, 7/21/2005Natural Language Processing45 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. El doctor recommendó una exploración del gato.
Lecture 1, 7/21/2005Natural Language Processing46 Overview N-grams Smoothing Backoff Caching Skipping Beyond N-grams Parsing Trigger Words
Lecture 1, 7/21/2005Natural Language Processing47 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, 7/21/2005Natural Language Processing48 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, 7/21/2005Natural Language Processing49 Computing the Probability of a Word Sequence P(w 1, …, w n ) = P(w 1 ).P(w 2 |w 1 ).P(w 3 |w 1,w 2 ). … P(w n |w 1, …,w n-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, 7/21/2005Natural Language Processing50 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
Lecture 1, 7/21/2005Natural Language Processing51 Using N-Grams For N-gram models P(w n-1,w n ) = P(w n | w n-1 ) P(w n-1 ) By the Chain Rule we can decompose a joint probability, e.g. P(w 1,w 2,w 3 )Chain Rule P(w 1,w 2,...,w n ) = P(w 1 |w 2,w 3,...,w n ) P(w 2 |w 3,...,w n ) … P(w n- 1 |w n ) P(w n ) 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| )
Lecture 1, 7/21/2005Natural Language Processing52 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”)
Lecture 1, 7/21/2005Natural Language Processing53 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, 7/21/2005Natural Language Processing54 Unigram probabilities (1-gram) 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, 7/21/2005Natural Language Processing55 N-grams for Language Generation C. E. Shannon, ``A mathematical theory of communication,'' Bell System Technical Journal, vol. 27, pp and , July and October, 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, 7/21/2005Natural Language Processing56 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, 7/21/2005Natural Language Processing57 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, 7/21/2005Natural Language Processing58 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, 7/21/2005Natural Language Processing59 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, 7/21/2005Natural Language Processing60 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, 7/21/2005Natural Language Processing61 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, 7/21/2005Natural Language Processing62 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
Lecture 1, 7/21/2005Natural Language Processing63 Using N-Grams For N-gram models P(w n-1,w n ) = P(w n | w n-1 ) P(w n-1 ) By the Chain Rule we can decompose a joint probability, e.g. P(w 1,w 2,w 3 )Chain Rule P(w 1,w 2,...,w n ) = P(w 1 |w 2,w 3,...,w n ) P(w 2 |w 3,...,w n ) … P(w n- 1 |w n ) P(w n ) 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| )
Lecture 1, 7/21/2005Natural Language Processing64 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, 7/21/2005Natural Language Processing65 A Simple Example P(I want to each Chinese food) = P(I | ) P(want | I) P(to | want) P(eat | to) P(Chinese | eat) P(food | Chinese)
Lecture 1, 7/21/2005Natural Language Processing66 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, 7/21/2005Natural Language Processing67.01British 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 I’m.26To eat.04 Tell.01Want Thai.06 I’d.04Want some.25 I
Lecture 1, 7/21/2005Natural Language Processing68 P(I want to eat British food) = P(I| ) P(want|I) P(to|want) P(eat|to) P(British|eat) P(food|British) =.25*.32*.65*.26*.001*.60 = vs. I want to eat Chinese food = 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, 7/21/2005Natural Language Processing69 BERP Bigram Counts Lunch Food Chinese Eat To Want I lunchFoodChineseEatToWantI
Lecture 1, 7/21/2005Natural Language Processing70 BERP Bigram Probabilities Normalization: divide each row's counts by appropriate unigram counts for w n-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 LunchFoodChineseEatToWantI
Lecture 1, 7/21/2005Natural Language Processing71 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, 7/21/2005Natural Language Processing72 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, 7/21/2005Natural Language Processing73 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, 7/21/2005Natural Language Processing74 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, 7/21/2005Natural Language Processing75 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, 7/21/2005Natural Language Processing76 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, 7/21/2005Natural Language Processing77 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, 7/21/2005Natural Language Processing78 Smoothing Techniques Every n-gram training matrix is sparse, even for very large corpora (Zipf’s law)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, 7/21/2005Natural Language Processing79 Smoothing Techniques Every n-gram training matrix is sparse, even for very large corpora (Zipf’s law)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, 7/21/2005Natural Language Processing80 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(w n-1 w n ) + 1 Incr unigram count by vocabulary size c(w n-1 ) + V
Lecture 1, 7/21/2005Natural Language Processing81 Discount: ratio of new counts to old (e.g. add-one smoothing changes the BERP bigram (to|want) from 786 to 331 (d c =.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, 7/21/2005Natural Language Processing82 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) Witten-Bell Discounting
Lecture 1, 7/21/2005Natural Language Processing83 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, 7/21/2005Natural Language Processing84 Re-estimate amount of probability mass for zero (or low count) ngrams by looking at ngrams with higher counts Estimate E.g. N 0 ’s adjusted count is a function of the count of ngrams that occur once, N 1 Assumes: word bigrams follow a binomial distribution We know number of unseen bigrams (VxV-seen) Good-Turing Discounting
Lecture 1, 7/21/2005Natural Language Processing85 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, 7/21/2005Natural Language Processing86 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