Naive Bayesian Text Classifier Event Models

NAÏVE BAYESIAN
TEXT CLASSIFIER
EVENT MODELS
NOTES AND HOW TO USE THEM
Alex Lin
Senior Architect
Intelligent Mining
alin@intelligentmining.com

Outline
  Quick refresher for
  Naïve Bayesian Text Classifier
  Event Models
  Multinomial Event Model
  Multi-variate Bernoulli Event Model
  Performance characteristics
  Why are they important?

2

Naïve Bayes Text Classifier
  Supervised Learning
  Labeled data set for training to classify the
unlabeled data
  Easy to implement and highly scalable
  It is often the first thing to try

  Successful cases: Email spam filtering,
News article categorization, Product
classification, Social content categorization

3

Bayesian Theorem
Likelihood Class Prior

P(X |Y )P(Y )
P(Y | X) =
P(X)
Posterior Evidence

€  Generative learning algorithms model “Likelihood”
P(X|Y) and “Class Prior” P(Y) then make
prediction based on Bayes Theorem.
6

P(X |Y )P(Y )
P(Y | X) =
P(X)
Y ∈ {0,1} d = w1,w 6 ,...,w n

Apply Bayes’ Theorem: €

P(w1,w 6 ,...,w n |Y = 0)P(Y = 0)
€P(Y = 0 | w1,w 6 ,...,w n ) =
P(w1,w 6 ,...,w n )
P(w1,w 6 ,...,w n |Y = 1)P(Y = 1)
P(Y = 1 | w1,w 6 ,...,w n ) =
P(w1,w 6 ,...,w n )
€
€

€ 7
€

P(X |Y )P(Y )
P(Y | X) =
P(X)
Y ∈ {0,1} d = w1,w 6 ,...,w n

To find most likely class label for d: €
Argmax P(Y = c | w1,w 6 ,...,w n )
€ € c
P(w1,w 6 ,...,w n |Y = c)P(Y = c)
= Argmax
c P(w1,w 6 ,...,w n )
€ = Argmax P(w1,w 6 ,...,w n |Y = c)P(Y = c)
c

How do we estimate likelihood?
€ 8

€

Differences of Two Models
∏ P(w i |Y = c)
Multinomial Event model i∈n

tf wi ,c Term Freq. of wi in Class c
P(w i |Y = c) =
c €
Sum of all Term Freq. in Class c

Multi-variate Bernoulli Event model
€ df wi ,c Doc Freq. of wi in Class c
P(w i |Y = c) = # of Docs in Class c
Nc
when wi not exists in d
10
P(w i |Y = c) = 1− P(w i |Y = c)
€

Comparison of Two Models
tf wi ,c
Multinomial: P(w i |Y = c) =
c
Label w1 w2 w3 w4 … wn
Y=0 tfw1,0 tfw2,0 tfw2,0 tfw2,0 … tfwn,0 |c|
€
Y=1 tfw1,1 tfw2,1 tfw2,1 tfw2,1 … tfwn,1 |c|

df wi ,c
Multi-variate Bernoulli: P(w i |Y = c) =
Nc
Label w1 w2 w3 w4 … wn
Y=0 dfw1,0 dfw2,0 dfw3,0 dfw4,0 … dfwn,0 N0
€
Y=0 !dfw1,0 !dfw2,0 !dfw3,0 !dfw4,0 … !dfwn,0 N0
Y=1 dfw1,1 dfw2,1 dfw3,1 dfw4,1 … dfwn,1 N1 11

Y=1 !dfw1,1 !dfw2,1 !dfw3,1 !dfw4,1 … !dfwn,1 N1

Comparison of Two Models
tf wi ,c ∏ P(w |Y = c)
Multinomial: P(w i |Y = c) =
c i∈n
i

d = {w1,w 2 ,...,w n } n: # of words in d w n ∈ {1,2,3,..., V }
c €
Argmax ∑ log(P(w i |Y = c)) + log( )
€
c
i∈n D
€
df wi ,c
Multi-variate Bernoulli: P(w i |Y = c) =
Nc
d = {w1,w 2 ,...,w n } n: # of words in vocabulary |V| w n ∈ {0,1}
€ c
Argmax ∑ log(P(w i |Y = c) (1− P(w i |Y = c))
wi 1−w i
) + log( )
c
i∈n D 12

€

Multinomial
For each
word in
doc

w1 w2 w3 w4 .. wn w1 w2 w3 w4 .. w5

Y=0 Y=1

A% B%

13

Multi-variate Bernoulli
For each
word in
doc

Y=1 A%
W
Y=0 1-A%
When W does exists
in doc

Y=1 B%
W’
Y=0 1-B%
When W does not
exists in doc
14

Performance Characteristics
  Multinomial would perform better Multi-variate
Bernoulli in most text classification tasks, especially
when vocabulary size |V| >= 1K
  Multinomial perform better when handling data sets
that have large variance in document length
  Multivariate Bernoulli could have the advantage of
dense data set
  Non-text features could be added as additional
Bernoulli variables. However it should not be added to
the vocabulary in Multinomial model
15

Why are they interesting?
  Certaindata points are more suitable for one
event model than the other.
  Example:
  Web page text + “Domain” + “Author”
  Social content text + “Who”
  Product name / desc + “Brand”
  We can create a Naïve Bayes Classifier that
combines event models
  Most importantly, try both on your data set
16

Thank you
  Any question or comment?

17

Appendix
  Laplace Smoothing
  Generative vs Discriminative Learning
  Multinomial Event Model

  Multi-variate Bernoulli Event Model
  Notation

18

Laplace Smoothing

Multinomial:
tf wi ,c + 1
P(w i |Y = c) =
c+V

Multi-variate Bernoulli:

€ df wi ,c + 1
P(w i |Y = c) =
Nc + 2
19

€

Generative vs. Discriminative
Discriminative Learning Algorithm:

Try to learn P(Y | X) directly or try to map input X
to labels Y directly

Generative Learning Algorithm:

Try to model P(X |Y) and
€ P(Y ) first, then use
Bayes theorem to find out
P(X |Y )P(Y )
P(Y | X) =
€ P(X)
€ 20

Multinomial Event Model
tf wi ,c
P(w i |Y = c) =
c

d = {w1,w 2 ,...,w n } n: # of words in d w n ∈ {1,2,3,..., V }

= Argmax P(w1,w 6 ,...,w n |Y = c)P(Y = c)
c
€ €
c tf wi ,c
= Argmax ∑ log( ) + log( )
c
i∈n
c D

21

Multi-variate Bernoulli Event Model
df wi ,c
P(w i |Y = c) =
Nc

d = {w1,w 2 ,...,w n } n: # of words in vocabulary |V| w n ∈ {0,1}

= Argmax P(w1,w 6 ,...,w n |Y = c)P(Y = c)
c
€
df wi ,c
df
€ wi ,c 1−wi c
= Argmax ∑ log(( ) )(1− ( wi
) ) + log( )
c
i∈n
Nc Nc D

22

Notation
  d: a document
  w: a word in a document

  X: observed data attributes

  Y: Class Label

  |V|: number of terms in vocabulary

  |D|: number of docs in training set

  |c|: number of docs in class c

  tf: Term frequency

  df: document frequency

  !df:
23

References
  McCallum, A., Nigam, K.: A comparison of event models for Naive Bayes
text classiﬁcation. In: Learning for Text Categorization: Papers from the AAAI
Workshop, AAAI Press (1998) 41–48 Technical Report WS-98-05
  Metsis, V., Androutsopoulos, I., Paliouras, G.: Spam Filtering with Naive
Bayes – Which Naive Bayes (2006) Third Conference on Email and Anti-Spam
(CEAS)
  Schneider, K: On Word Frequency Information and Negative Evidence in
Naive Bayes Text Classification (2004) España for Natural Language
Processing, EsTAL

24

Naive Bayesian Text Classifier Event Models

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