naive bayes presentation

Naive Bayes

Md Enamul Haque Chowdhury

ID : CSE013083972D

University of Luxembourg

(Based on Ke Chen and Ashraf Uddin Presentation)

Contents

Background

Bayes Theorem

Bayesian Classifier

Naive Bayes

Uses of Naive Bayes classification

Relevant Issues

Advantages and Disadvantages

Some NBC Applications

Conclusions

1

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

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

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.

Note that we can drop P(D) as the probability of the data is constant (and

independent of the hypothesis).

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4

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:

This hypothesis is called the maximum likelihood hypothesis.

)|(maxarg hDPhHh

ML

5

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

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

Naive Bayes

Bayes classification

Difficulty: learning the joint probability

Naive Bayes classification

-Assumption that all input features are conditionally independent!

-MAP classification rule: for

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8

Naive Bayes

Algorithm: Discrete-Valued Features

-Learning Phase: Given a training set S,

Output: conditional probability tables; for elements

-Test Phase: Given an unknown instance

Look up tables to assign the label c* to X´ if

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9

Example

10

Example

Learning Phase :

P(Play=Yes) = 9/14

P(Play=No) = 5/14

Outlook Play=Yes Play=No

Sunny 2/9 3/5Overcast 4/9 0/5Rain 3/9 2/5

Temperature Play=Yes Play=No

Hot 2/9 2/5Mild 4/9 2/5Cool 3/9 1/5

Humidity Play=Yes Play=No

High 3/9 4/5Normal 6/9 1/5

Wind Play=Yes Play=No

Strong 3/9 3/5Weak 6/9 2/5

11

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

-Decision making with the MAP rule:

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

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

Naive Bayes

Algorithm: Continuous-valued Features

- Numberless values for a feature

- Conditional probability often modeled with the normal distribution

- 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

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13

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

-Learning Phase: output two Gaussian models for P(temp|C)

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14

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

Why text classification?

Learning which articles are of interest

Classify web pages by topic

Information extraction

Internet filters

16

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

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

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

Relevant Issues

Violation of Independence Assumption

Zero conditional probability Problem

20

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

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

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

Naive Bayes Laplace Correction

To eliminate zeros, we use add-one or Laplace smoothing, which simply adds one to

each count

24

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

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

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

Some NBC Applications

Credit scoring

Marketing applications

Employee selection

Image processing

Speech recognition

Search engines…

28

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

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

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Questions ?