COMP24111 Machine Learning

Naïve Bayes Classifier

Ke Chen

COMP24111 Machine Learning2

Outline

• Background

• Probability Basics

• Probabilistic Classification

• Naïve Bayes – Principle and Algorithms

– Example: Play Tennis

• Relevant Issues

• Summary

COMP24111 Machine Learning3

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

COMP24111 Machine Learning4

Probability Basics

• Prior, conditional and joint probability for random variables– Prior probability:

– Conditional probability:

– Joint probability:

– Relationship:

– Independence:

• Bayesian Rule

)| ,)( 121 XP(XX|XP 2

)()()(

)(X

XX

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|CP

)(XP

) )( ),,( 22 ,XP(XPXX 11 XX

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)()() ),()|( ),()|( 212121212 XPXP,XP(XXPXXPXPXXP 1

EvidencePriorLikelihood

Posterior

COMP24111 Machine Learning5

Probability Basics

• Quiz: We have two six-sided dice. When they are tolled, it could end up with the following occurance: (A) dice 1 lands on side “3”, (B) dice 2 lands on side “1”, and (C) Two dice sum to eight. Answer the following questions:

? to equal ),( Is 8)

?),( 7)

?),( 6)

?)|( 5)

?)|( 4)

? 3)

? 2)

? )( )1

P(C)P(A)CAP

CAP

BAP

ACP

BAP

P(C)

P(B)

AP

COMP24111 Machine Learning6

Probabilistic Classification

• Establishing a probabilistic model for classification– Discriminative model

),, , )( 1 n1L X(Xc,,cC|CP XX

),,,( 21 nxxx x

Discriminative Probabilistic Classifier

1x 2x nx

)|( 1 xcP )|( 2 xcP )|( xLcP

COMP24111 Machine Learning7

Probabilistic Classification

• Establishing a probabilistic model for classification (cont.)– Generative model

),, , )( 1 n1L X(Xc,,cCC|P XX

GenerativeProbabilistic Model

for Class 1

)|( 1cP x

1x 2x nx

GenerativeProbabilistic Model

for Class 2

)|( 2cP x

1x 2x nx

GenerativeProbabilistic Model

for Class L

)|( LcP x

1x 2x nx

),,,( 21 nxxx x

COMP24111 Machine Learning8

Probabilistic Classification

• MAP classification rule

– MAP: Maximum A Posterior

– Assign x to c* if

• Generative classification with the MAP rule– Apply Bayesian rule to convert them into posterior

probabilities

– Then apply the MAP rule

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PcCPcC|P

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ii

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xXxX

xX

COMP24111 Machine Learning9

Naïve Bayes

• Bayes classification

Difficulty: learning the joint probability

• Naïve Bayes classification– Assumption that all input features are conditionally

independent!

– MAP classification rule: for

)()|,,()()( )( 1 CPCXXPCPC|P|CP n XX

)|,,( 1 CXXP n

)|()|()|(

)|,,()|(

)|,,(),,,|()|,,,(

21

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),,,( 21 nxxx x

COMP24111 Machine Learning10

Naïve Bayes

;in examples with )|( estimate)|(ˆ

),1 ;,,1( featureeach of valuefeatureevery For

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S

S

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1

),,( 1 naa X

COMP24111 Machine Learning11

Example

• Example: Play Tennis

COMP24111 Machine Learning12

Example

• Learning Phase

Outlook Play=Yes

Play=No

Sunny 2/9 3/5Overcas

t4/9 0/5

Rain 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

P(Play=Yes) = 9/14P(Play=No) = 5/14

COMP24111 Machine Learning13

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=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(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(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”.

COMP24111 Machine Learning14

Naïve 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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cC

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C which for examplesof values featureof (avearage) mean :

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)|(̂ 2

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Ln ccCXX ,, ),,,( for 11 XLn LicCP i ,,1 )(

),,( 1 naa X

COMP24111 Machine Learning15

Naïve 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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2

2

2

2

2

2

xxNoxP

xxYesxP

COMP24111 Machine Learning16

Relevant Issues

• Violation of Independence Assumption– For many real world tasks,

– Nevertheless, naïve Bayes works surprisingly well anyway!

• Zero conditional probability Problem– If no example contains the feature value

– In this circumstance, during test

– For a remedy, conditional probabilities re-estimated with

)|()|( )|,,( 11 CXPCXPCXXP nn

0)|(̂ , ijkjjkj cCaXPaX

0)|(̂)|(̂)|(̂ 1 inijki cxPcaPcxP

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) of values possible for /1 (usually, estimate prior :

whichfor examples training of number :

C and whichfor examples training of number :

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COMP24111 Machine Learning17

Summary

• Naïve Bayes: the conditional independence assumption– Training is very easy and fast; just requiring considering each

attribute in each class separately

– Test is straightforward; just looking up tables or calculating conditional probabilities with estimated distributions

• A popular generative model– Performance competitive to most of state-of-the-art classifiers

even in presence of violating independence assumption

– Many successful applications, e.g., spam mail filtering

– A good candidate of a base learner in ensemble learning

– Apart from classification, naïve Bayes can do more…