This document provides an overview of Naive Bayes classification. It begins with background on classification methods, then covers Bayes' theorem and how it relates to Bayesian and maximum likelihood classification. The document introduces Naive Bayes classification, which makes a strong independence assumption to simplify probability calculations. It discusses algorithms for discrete and continuous features, and addresses common issues like dealing with zero probabilities. The document concludes by outlining some applications of Naive Bayes classification and its advantages of simplicity and effectiveness for many problems.

1. Naive Bayes
Md Enamul Haque Chowdhury
ID : CSE013083972D
University of Luxembourg
(Based on Ke Chen and Ashraf Uddin Presentation)

2. Contents
 Background
 Bayes Theorem
 Bayesian Classifier
 Naive Bayes
 Uses of Naive Bayes classification
 Relevant Issues
 Advantages and Disadvantages
 Some NBC Applications
 Conclusions
1

3. Background
 There are three methods to establish a classifier
a) Model a classification rule directly
Examples: k-NN, decision trees, perceptron, SVM
b) Model the probability of class memberships given input data
Example: perceptron with the cross-entropy cost
c) Make a probabilistic model of data within each class
Examples: Naive Bayes, Model based classifiers
 a) and b) are examples of discriminative classification
 c) is an example of generative classification
 b) and c) are both examples of probabilistic classification
2

4. Bayes Theorem
 Given a hypothesis h and data D which bears on the hypothesis:
 P(h): independent probability of h: prior probability
 P(D): independent probability of D
 P(D|h): conditional probability of D given h: likelihood
 P(h|D): conditional probability of h given D: posterior probability
3

5. Maximum A Posterior
 Based on Bayes Theorem, we can compute the Maximum A Posterior (MAP)
hypothesis for the data
 We are interested in the best hypothesis for some space H given observed training
data D.
H: set of all hypothesis.
h argmaxP(h | D)
h H
MAP


P D h P h
( | ) ( )
P D
( )
argmax
hH

argmaxP(D| h)P(h)
hH

Note that we can drop P(D) as the probability of the data is constant (and
independent of the hypothesis).
4

6. Maximum Likelihood
 Now assume that all hypothesis are equally probable a prior, i.e. P(hi ) = P(hj ) for all
hi, hj belong to H.
 This is called assuming a uniform prior. It simplifies computing the posterior:
h argmaxP(D| h)
h H
ML


 This hypothesis is called the maximum likelihood hypothesis.
5

7. Bayesian Classifier
 The classification problem may be formalized using a-posterior probabilities:
 P(C|X) = prob. that the sample tuple X=<x1,…,xk> is of class C.
 E.g. P(class=N | outlook= sunny, windy=true,…)
 Idea: assign to sample X the class label C such that P(C|X) is maximal
6

8. Estimating a-posterior probabilities
 Bayes theorem:
P(C|X) = P(X|C)·P(C) / P(X)
 P(X) is constant for all classes
 P(C) = relative freq of class C samples
 C such that P(C|X) is maximum = C such that P(X|C)·P(C) is maximum
 Problem: computing P(X|C) is unfeasible!
7

9. Naive Bayes
 Bayes classification
( ) ( ) ( ) ( , , | ) ( ) 1 P C| P |C P C P X X C P C n X  X  
Difficulty: learning the joint probability
 Naive Bayes classification
-Assumption that all input features are conditionally independent!
P X X X C P X X X C P X X C
( , ,  , | )  ( | ,  , , ) ( , 
, | )
n n n
1 2 1 2 2
-MAP classification rule: for
P X C P X X C
 
( | ) ( , , | )
1 2
P X C P X C P X C
( | ) ( | ) ( | )
1 2
n
n
 
( , , , ) 1 2 n x  x x  x
*
[P(x | c ) P(x | c )]P(c ) [P(x | c) P(x | c)]P(c), c c , c c , ,c n 1
n 1
L * * *
1            
8

10. Naive Bayes
 Algorithm: Discrete-Valued Features
-Learning Phase: Given a training set S,
c (c c , ,c )
For each target value of 1
   
i i L
ˆ ( ) estimate ( ) with examples in ;
P C  c  P C 
c
i i
x X j n k ,N
For every feature value of each feature (  1,    , ;  1,   
)
jk j j
ˆ ( | ) estimate ( | ) with examples in ;
P X  x C  c  P X  x C 
c
X N L j j , 
Output: conditional probability tables; for elements
-Test Phase: Given an unknown instance
( , , ) 1 n X  a    a
Look up tables to assign the label c* to X´ if
S
S
j jk i j jk i
[Pˆ(a  | c * )  Pˆ(a  | c * )]Pˆ(c *
)  [Pˆ(a  | c)  Pˆ(a  | c)]Pˆ(c), c  c *
, c  c , 
,c 1 n 1
n 1
L 9

11. Example
10

12. Example
Learning Phase :
Outlook Play=Yes Play=No
Sunny 2/9 3/5
Overcast 4/9 0/5
Rain 3/9 2/5
P(Play=Yes) = 9/14
P(Play=No) = 5/14
Temperature Play=Yes Play=No
Hot 2/9 2/5
Mild 4/9 2/5
Cool 3/9 1/5
Humidity Play=Yes Play=No
High 3/9 4/5
Normal 6/9 1/5
Wind Play=Yes Play=No
Strong 3/9 3/5
Weak 6/9 2/5
11

13. Example
 Test Phase :
-Given a new instance, predict its label
x´=(Outlook=Sunny, Temperature=Cool, Humidity=High, Wind=Strong)
-Look up tables achieved in the learning phrase
P(Outlook=Sunny|Play=Yes) = 2/9
P(Temperature=Cool|Play=Yes) = 3/9
P(Huminity=High|Play=Yes) = 3/9
P(Wind=Strong|Play=Yes) = 3/9
P(Play=Yes) = 9/14
-Decision making with the MAP rule:
P(Outlook=Sunny|Play=No) = 3/5
P(Temperature=Cool|Play==No) = 1/5
P(Huminity=High|Play=No) = 4/5
P(Wind=Strong|Play=No) = 3/5
P(Play=No) = 5/14
P(Yes|x´): [ P(Sunny|Yes) P(Cool|Yes) P(High|Yes) P(Strong|Yes) ] P(Play=Yes) = 0.0053
P(No|x´): [ P(Sunny|No) P(Cool|No) P(High|No) P(Strong|No) ] P(Play=No) = 0.0206
Given the fact P(Yes|x´) < P(No|x´) , we label x´ to be “No”.
12

14. Naive Bayes
 Algorithm: Continuous-valued Features
- Numberless values for a feature
- Conditional probability often modeled with the normal distribution
 
( 
)
ˆ ( | ) 2
j ji

1
  
2
exp
2
X c
2
: mean (avearage) of feature values of examples for whichC

ji j i
ji j i
- Learning Phase:
Output: normal distributions and
- Test Phase: Given an unknown instance
-Instead of looking-up tables, calculate conditional probabilities with all the normal
distributions achieved in the learning phrase
-Apply the MAP rule to make a decision
ji
ji
j i
C c
X
P X C c








: standard deviation of feature values X of examples for which



n L for (X , , X ), C c , ,c 1 1 X        
P C c i L i nL (  ) 1,  ,
( , , ) 1 n X  a    a
13

15. Naive Bayes
 Example: Continuous-valued Features
-Temperature is naturally of continuous value.
Yes: 25.2, 19.3, 18.5, 21.7, 20.1, 24.3, 22.8, 23.1, 19.8
No: 27.3, 30.1, 17.4, 29.5, 15.1
-Estimate mean and variance for each class
N
N
1
2 2
 
 
n x
x
1
  ,
  

N n
1
( )
n
n
N
1
 
 21.64, 
2.35
 
Yes Yes
 
23.88, 7.09
No No
-Learning Phase: output two Gaussian models for P(temp|C)


 

 
1
 
( 21.64)
 
 
1

 

 

 

( 23.88)
50.25
exp
7.09 2
ˆ ( | )
11.09
exp
2.35 2
ˆ ( | )
2
2
x
P x No
x
P x Yes

14

16. Uses of Naive Bayes classification
 Text Classification
 Spam Filtering
 Hybrid Recommender System
- Recommender Systems apply machine learning and data mining techniques for
filtering unseen information and can predict whether a user would like a given
resource
 Online Application
- Simple Emotion Modeling
15

17. Why text classification?
 Learning which articles are of interest
 Classify web pages by topic
 Information extraction
 Internet filters
16

18. Examples of Text Classification
 CLASSES=BINARY
 “spam” / “not spam”
 CLASSES =TOPICS
 “finance” / “sports” / “politics”
 CLASSES =OPINION
 “like” / “hate” / “neutral”
 CLASSES =TOPICS
 “AI” / “Theory” / “Graphics”
 CLASSES =AUTHOR
 “Shakespeare” / “Marlowe” / “Ben Jonson”
17

19. Naive Bayes Approach
 Build the Vocabulary as the list of all distinct words that appear in all the documents
of the training set.
 Remove stop words and markings
 The words in the vocabulary become the attributes, assuming that classification is
independent of the positions of the words
 Each document in the training set becomes a record with frequencies for each word
in the Vocabulary.
 Train the classifier based on the training data set, by computing the prior probabilities
for each class and attributes.
 Evaluate the results on Test data
18

20. Text Classification Algorithm: Naive Bayes
 Tct – Number of particular word in particular class
 Tct’ – Number of total words in particular class
 B´ – Number of distinct words in all class
19

21. Relevant Issues
 Violation of Independence Assumption
 Zero conditional probability Problem
20

22. Violation of Independence Assumption
 Naive Bayesian classifiers assume that the effect of an attribute value on a given
class is independent of the values of the other attributes. This assumption is called
class conditional independence. It is made to simplify the computations involved and,
in this sense, is considered “naive.”
21

23. Improvement
 Bayesian belief network are graphical models, which unlike naive Bayesian
classifiers, allow the representation of dependencies among subsets of attributes.
 Bayesian belief networks can also be used for classification.
22

24. Zero conditional probability Problem
 If a given class and feature value never occur together in the training set then the
frequency-based probability estimate will be zero.
 This is problematic since it will wipe out all information in the other probabilities when
they are multiplied.
 It is therefore often desirable to incorporate a small-sample correction in all
probability estimates such that no probability is ever set to be exactly zero.
23

25. Naive Bayes Laplace Correction
 To eliminate zeros, we use add-one or Laplace smoothing, which simply adds one to
each count
24

26. Example
 Suppose that for the class buys computer D (yes) in some training database, D, containing 1000
tuples.
 we have 0 tuples with income D low,
 990 tuples with income D medium, and
 10 tuples with income D high.
 The probabilities of these events, without the Laplacian correction, are 0, 0.990 (from 990/1000),
and 0.010 (from 10/1000), respectively.
 Using the Laplacian correction for the three quantities, we pretend that we have 1 more tuple for
each income-value pair. In this way, we instead obtain the following probabilities :
respectively. The “corrected” probability estimates are close to their “uncorrected” counterparts,
yet the zero probability value is avoided.
25

27. Advantages
• Advantages :
 Easy to implement
 Requires a small amount of training data to estimate the parameters
 Good results obtained in most of the cases
26

28. Disadvantages
 Disadvantages:
 Assumption: class conditional independence, therefore loss of accuracy
 Practically, dependencies exist among variables
-E.g., hospitals: patients: Profile: age, family history, etc.
Symptoms: fever, cough etc., Disease: lung cancer, diabetes, etc.
 Dependencies among these cannot be modelled by Naïve Bayesian Classifier
27

29. Some NBC Applications
 Credit scoring
 Marketing applications
 Employee selection
 Image processing
 Speech recognition
 Search engines…
28

30. Conclusions
 Naive Bayes is:
- Really easy to implement and often works well
- Often a good first thing to try
- Commonly used as a “punching bag” for smarter algorithms
29

31. References
 http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/mlbook/ch6.pdf
 Data Mining: Concepts and Techniques, 3rd
Edition, Han & kamber & Pei ISBN: 9780123814791
 http://en.wikipedia.org/wiki/Naive_Bayes_classifier
 http://www.slideshare.net/ashrafmath/naive-bayes-15644818
 http://www.slideshare.net/gladysCJ/lesson-71-naive-bayes-classifier
30

32. Questions ?