Transcript Language Models

Language Modeling
Page 1
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?
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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
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
•
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The, of, to, in, and, for, that, is, on, it, by, with, as, at, said, mister, from, its, are, he,
million
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?
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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...
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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
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Wall Street Journal
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AP news
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DARPA/NIST text/speech corpora (Call Home, ATIS, switchboard,
Broadcast News, TDT, Communicator)
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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
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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
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P(wn |w1n1)
by
P(wn |wn 1)
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
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trigrams, 4-grams, …
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the higher n is, the more data needed to train
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backoff models
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Using N-Grams
For N-gram models
P(wn |w1n1)
•
P(wn |wnn1N 1)

•
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>)
n
P(w )   P(wk | wk 1)
n
1
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k 1
Training and Testing
N-Gram probabilities come from a training corpus
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overly narrow corpus: probabilities don't generalize
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overly general corpus: probabilities don't reflect task or domain
A separate test corpus is used to evaluate the model, typically using
standard metrics
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held out test set; development test set
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cross validation
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results tested for statistical significance
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A Simple Example
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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
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
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<start> I
<start> I’d
<start> Tell
<start> I’m
I want
I would
I don’t
I have
Want to
.25
.06
.04
.02
.32
.29
.08
.04
.65
Want some
Want Thai
To eat
To have
To spend
To be
British food
British restaurant
British cuisine
.04
.01
.26
.14
.09
.02
.60
.15
.01
Want a
.05
British lunch
.01
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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
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
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BERP Bigram Probabilities
Normalization: divide each row's counts by appropriate unigram counts for wn-1
I
Want
To
Eat
Chinese
Food
Lunch
3437
1215
3256
938
213
1506
459
Computing the bigram probability of I I
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C(I,I)/C(all I)
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p (I|I) = 8 / 3437 = .0023
Maximum Likelihood Estimation (MLE): relative frequency of e.g.
freq(w1, w2)
freq(w1)
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What do we learn about the language?
What's being captured with ...
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P(want | I) = .32
•
P(to | want) = .65
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P(eat | to) = .26
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P(food | Chinese) = .56
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P(lunch | eat) = .055
What about...
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P(I | I) = .0023
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P(I | want) = .0025
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P(I | food) = .013
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P(I | I) = .0023 I I I I want
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P(I | want) = .0025 I want I want
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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...
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Every enter now severally so, let
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Hill he late speaks; or! a more to leg less first you enter
With bigrams...
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What means, sir. I confess she? then all sorts, he is trim, captain.
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Why dost stand forth thy canopy, forsooth; he is this palpable hit the King
Henry.
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Trigrams
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Sweet prince, Falstaff shall die.
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This shall forbid it should be branded, if renown made it empty.
Quadrigrams
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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 ngrams 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 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



•
c 1 N
N V
i
Normalize by N to get the new unigram probability:
p*  c 1
i N V
i
C ( xyz)  1
P( z | xy) 
C ( xy)  V
Add delta smoothing
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C ( xyz)  
P( z | xy) 
C ( xy)  V
Witten-Bell Discounting
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
T
N T
This probability mass will assigned to unseen cases
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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)
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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
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T
N T
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
Number of bigram types
starting with wx
1
T ( wx )
P( wi | wx ) 
Z ( wx ) N ( wx )  T ( wx )
Number of bigrams
starting with wx
that were not seen
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Actual frequency of
bigrams beginning
with wx
Good-Turing Discounting
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?
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N
c*  c  1 c 1
Nc
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:
–
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Where trigram unavailable back off to bigram if available, o.w. unigram
probability
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 40
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
CORPUS
100 documents
Each sentence
scored (1 to 20)
Machine
Learning
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CorpusBased
Model
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
X  {( x1 , y1 ), ( x2 , y2 )...( x( n 1) x100 , y( n 1) x100 ), ( xnx100 , ynx100 )}
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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:
Remp
1

N
N
 L( y , f ( x ))
i
i
i 1
L( yi , f ( xi ))  ( yi  f ( xi ))
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2
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
f ( x; )  1 x  0
y  mx  c
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
12
The line we have found by
minimizing average squared
(xi, f(xi))
error is our model
(summarizer)
Score Error (yi - f(xi))
10
8
6
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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Series1
(xi, yi)
4
2
0
0
1
2
3
Sentence position
4
5
6
7
8
9
10
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
All figures in the given example are from electronic textbook StatSoft
•
•
•
•
Page 49
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)
Example (RED or GREEN classification)
Prior probability of GREEN  # of GREEN objects
Total # of objects
Prior probability of RED 
# of RED objects
Total # of objects
There are total of 60 objects with 40 GREEN and 20 RED
Prior probability of GREEN  40
60
Prior probability of RED 
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20
60
Example (RED or GREEN classification)
Need to figure out if the
test object is GREEN or
RED
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
Likelihood of X being GREEN 
# of GREEN objects in vicinity of X
Total # of GREEN
Likelihood of X being RED 
# of RED objects in vicinity of X
Total # of RED
Page 51
Example (RED or GREEN classification)
Need to figure out if the
test object is GREEN or
RED
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
Likelihood of X being GREEN 
# of GREEN objects in vicinity of X  1/40
Total # of GREEN
Likelihood of X being RED 
# of RED objects in vicinity of X  3/20
Total # of RED
Page 52
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 our
Bayesian 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.
Page 55
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
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 SHOWING
1 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
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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
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
Length
Noun Count
Verb Count
# Person
Name
# Place Name
# Orgzn
Name
.
.
.
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780
45
3
4
7
67
23
420
12
56
0
7
3
4
.
.
.
.
.
.
HIST
PHYSICS
…
940
36
3
8
6
55
21
.
.
.
HIST
Feature Vectors
Length
Noun Count
Verb Count
# Person
Name
# Place Name
# Orgzn
Name
.
.
.
780
45
3
4
7
67
23
420
12
56
0
7
3
4
.
.
.
.
.
.
HIST
PHYSICS
…
940
36
3
8
6
55
21
940
36
3
8
6
55
21
.
.
.
.
.
.
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
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HIST