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LING 581: Advanced Computational Linguistics
Lecture Notes
January 12th
Course
• Webpage for lecture slides– http://dingo.sbs.arizona.edu/~sandiway/ling581-
12/
• Meeting information
Course Objectives
• Follow-on course to LING/C SC/PSYC 438/538 Computational Linguistics:– continue with J&M: 25 chapters, a lot of material not covered
in 438/538
• And gain project experience– dealing with natural language software packages– Installation, input data formatting– operation – project exercises– useful “real-world” computational experience
– abilities gained will be of value to employers
Computational Facilities
• Advise using your own laptop/desktop– we can also make use of this computer lab
• but you don’t have installation rights on these computers
• PlatformsWindows is possible but you really should run some variant of Unix…
(your task #1 for this week)– Linux (separate bootable partition or via virtualization software)
• de facto standard for advanced/research software • https://www.virtualbox.org/ (free!)
– Cygwin on Windows• http://www.cygwin.com/• Linux-like environment for Windows making it possible to port software running on POSIX
systems (such as Linux, BSD, and Unix systems) to Windows.
– MacOS X• Not quite Linux, some porting issues, especially with C programs, can use Virtual Box
Grading
• Completion of all homework task and course project will result in a satisfactory grade (A)
Homework Task 1: Install Tregex
Runs in java
Last semester
• 438/538: Language models
Language Models and N-grams
• given a word sequence– w1 w2 w3 ... wn
• chain rule– how to compute the probability of a sequence of words– p(w1 w2) = p(w1) p(w2|w1) – p(w1 w2 w3) = p(w1) p(w2|w1) p(w3|w1w2) – ...– p(w1 w2 w3...wn) = p(w1) p(w2|w1) p(w3|w1w2)... p(wn|w1...wn-2 wn-1)
• note– It’s not easy to collect (meaningful) statistics on p(wn|wn-1wn-2...w1)
for all possible word sequences
Language Models and N-grams
• Given a word sequence– w1 w2 w3 ... wn
• Bigram approximation– just look at the previous word only (not all the proceedings words) – Markov Assumption: finite length history– 1st order Markov Model– p(w1 w2 w3...wn) = p(w1) p(w2|w1) p(w3|w1w2) ...p(wn|w1...wn-3wn-2wn-1)
– p(w1 w2 w3...wn) p(w1) p(w2|w1) p(w3|w2)...p(wn|wn-1)
• note– p(wn|wn-1) is a lot easier to collect data for (and thus estimate well) than p(wn|
w1...wn-2 wn-1)
Language Models and N-grams
• Trigram approximation – 2nd order Markov Model– just look at the preceding two words only– p(w1 w2 w3 w4...wn) = p(w1) p(w2|w1) p(w3|w1w2) p(w4|w1w2w3)...p(wn|
w1...wn-3wn-2wn-1)
– p(w1 w2 w3...wn) p(w1) p(w2|w1) p(w3|w1w2)p(w4|w2w3)...p(wn |wn-2
wn-1)
• note– p(wn|wn-2wn-1) is a lot easier to estimate well than p(wn|w1...wn-2 wn-1)
but harder than p(wn|wn-1 )
Language Models and N-grams
• estimating from corpora– how to compute bigram probabilities– p(wn|wn-1) = f(wn-1wn)/f(wn-1w) w is any word
– Since f(wn-1w) = f(wn-1) f(wn-1) = unigram frequency for wn-1
– p(wn|wn-1) = f(wn-1wn)/f(wn-1) relative frequency
• Note:– The technique of estimating (true) probabilities using a
relative frequency measure over a training corpus is known as maximum likelihood estimation (MLE)
Motivation for smoothing
• Smoothing: avoid zero probability estimates• Consider
• what happens when any individual probability component is zero?– Arithmetic multiplication law: 0×X = 0– very brittle!
• even in a very large corpus, many possible n-grams over vocabulary space will have zero frequency– particularly so for larger n-grams
p(w1 w2 w3...wn) p(w1) p(w2|w1) p(w3|w2)...p(wn|wn-1)
Language Models and N-grams
• Example:
unigram frequencies
wn-1wn bigram frequencies
bigram probabilities
sparse matrix
zeros render probabilities unusable
(we’ll need to add fudge factors - i.e. do smoothing)
wn-1
wn
Smoothing and N-grams
• sparse dataset means zeros are a problem– Zero probabilities are a problem
• p(w1 w2 w3...wn) p(w1) p(w2|w1) p(w3|w2)...p(wn|wn-1) bigram model
• one zero and the whole product is zero– Zero frequencies are a problem
• p(wn|wn-1) = f(wn-1wn)/f(wn-1) relative frequency
• bigram f(wn-1wn) doesn’t exist in dataset
• smoothing– refers to ways of assigning zero probability n-grams a non-zero value
Smoothing and N-grams
• Add-One Smoothing (4.5.1 Laplace Smoothing)– add 1 to all frequency counts– simple and no more zeros (but there are better methods)
• unigram– p(w) = f(w)/N (before Add-One)
• N = size of corpus– p(w) = (f(w)+1)/(N+V) (with Add-One)– f*(w) = (f(w)+1)*N/(N+V) (with Add-One)
• V = number of distinct words in corpus• N/(N+V) normalization factor adjusting for the effective increase in the corpus
size caused by Add-One• bigram
– p(wn|wn-1) = f(wn-1wn)/f(wn-1) (before Add-One)– p(wn|wn-1) = (f(wn-1wn)+1)/(f(wn-1)+V) (after Add-One)– f*(wn-1 wn) = (f(wn-1 wn)+1)* f(wn-1) /(f(wn-1)+V) (after Add-One)
must rescale so that total probability mass stays at 1
Smoothing and N-grams
• Add-One Smoothing– add 1 to all frequency counts
• bigram– p(wn|wn-1) = (f(wn-1wn)+1)/(f(wn-1)+V) – (f(wn-1 wn)+1)* f(wn-1) /(f(wn-1)+V)
• frequencies
Remarks:perturbation problem
add-one causes largechanges in somefrequencies due to relative size of V (1616)
want to: 786 338
= figure 6.8
= figure 6.4
Smoothing and N-grams
• Add-One Smoothing– add 1 to all frequency counts
• bigram– p(wn|wn-1) = (f(wn-1wn)+1)/(f(wn-1)+V) – (f(wn-1 wn)+1)* f(wn-1) /(f(wn-1)+V)
• Probabilities
Remarks:perturbation problem
similar changes inprobabilities
= figure 6.5
= figure 6.7
Smoothing and N-grams
• let’s illustrate the problem– take the bigram case:– wn-1wn
– p(wn|wn-1) = f(wn-1wn)/f(wn-1)
– suppose there are cases– wn-1w0
1 that don’t occur in the corpus
probability mass
f(wn-1)
f(wn-1wn)
f(wn-1w01)=0
f(wn-1w0m)=0
...
Smoothing and N-grams
• add-one– “give everyone 1”
probability mass
f(wn-1)
f(wn-1wn)+1
f(wn-1w01)=1
f(wn-1w0m)=1
...
Smoothing and N-grams
• add-one– “give everyone 1”
probability mass
f(wn-1)
f(wn-1wn)+1
f(wn-1w01)=1
f(wn-1w0m)=1
... V = |{wi)|
• redistribution of probability mass
– p(wn|wn-1) = (f(wn-1wn)+1)/(f(wn-1)+V)
Smoothing and N-grams
• Excel spreadsheet available– addone.xls
Smoothing and N-grams
• Good-Turing Discounting (4.5.2)– Nc = number of things (= n-grams) that occur c times in the corpus– N = total number of things seen– Formula: smoothed c for Nc given by c* = (c+1)Nc+1/Nc
– Idea: use frequency of things seen once to estimate frequency of things we haven’t seen yet– estimate N0 in terms of N1…– Formula: P*(things with zero freq) = N1/N– smaller impact than Add-One
• Textbook Example:– Fishing in lake with 8 species – Bass, carp, catfish, eel, perch, salmon, trout, whitefish– Sample data:– 10 carp, 3 perch, 2 whitefish, 1 trout, 1 salmon, 1 eel
– P(new fish unseen) = 3/18– P(next fish=trout) = 1/18 (but, we have reassigned probability mass…)– C*(trout) = 2*N2/N1=2(1/3)=0.67 (discounted from 1)
Language Models and N-grams
• N-gram models– they’re technically easy to compute
• (in the sense that lots of training data are available)
– but just how good are these n-gram language models?
– and what can they show us about language?
Language Models and N-grams
approximating Shakespeare– generate random sentences using n-grams– train on complete Works of Shakespeare
• Unigram (pick random, unconnected words)
• Bigram
Language Models and N-grams
• Approximating Shakespeare (section 6.2)– generate random sentences using n-grams– train on complete Works of Shakespeare
• Trigram
• Quadrigram
Remarks:dataset size problem
training set is small884,647 words29,066 different words
29,0662 = 844,832,356 possible bigrams
for the random sentence generator, this means very limited choices forpossible continuations,which means program can’t be very innovative for higher n
Language Models and N-grams
• Aside: http://hemispheresmagazine.com/contests/2004/intro.htm
Language Models and N-grams
• N-gram models + smoothing– one consequence of smoothing is that– every possible concatentation or sequence
of words has a non-zero probability
Colorless green ideas
• examples– (1) colorless green ideas sleep furiously– (2) furiously sleep ideas green colorless
• Chomsky (1957):– . . . It is fair to assume that neither sentence (1) nor
(2) (nor indeed any part of these sentences) has ever occurred in an English discourse. Hence, in any statistical model for grammaticalness, these sentences will be ruled out on identical grounds as equally `remote' from English. Yet (1), though nonsensical, is grammatical, while (2) is not.
• idea– (1) is syntactically valid, (2) is word salad
• Statistical Experiment (Pereira 2002)
Colorless green ideas
• examples– (1) colorless green ideas sleep furiously– (2) furiously sleep ideas green colorless
• Statistical Experiment (Pereira 2002) wiwi-1
bigram language model
• example– colorless green ideas sleep furiously
• First hit
Interesting things to Google
Interesting things to Google
• example– colorless green ideas sleep furiously
• first hit– compositional semantics– a green idea is, according to well established usage of the word "green" is
one that is an idea that is new and untried. – again, a colorless idea is one without vividness, dull and unexciting. – so it follows that a colorless green idea is a new, untried idea that is
without vividness, dull and unexciting. – to sleep is, among other things, is to be in a state of dormancy or inactivity,
or in a state of unconsciousness. – to sleep furiously may seem a puzzling turn of phrase but one reflects that
the mind in sleep often indeed moves furiously with ideas and images flickering in and out.
Interesting things to Google
• example– colorless green ideas sleep furiously
• another hit: (a story)– "So this is our ranking system," said Chomsky. "As you can see, the
highest rank is yellow." – "And the new ideas?" – "The green ones? Oh, the green ones don't get a color until
they've had some seasoning. These ones, anyway, are still too angry. Even when they're asleep, they're furious. We've had to kick them out of the dormitories - they're just unmanageable."
– "So where are they?" – "Look," said Chomsky, and pointed out of the window. There below,
on the lawn, the colorless green ideas slept, furiously.
