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
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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:
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COMP24111 Machine Learning6
Probabilistic Classification
• Establishing a probabilistic model for classification– Discriminative model
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Discriminative Probabilistic Classifier
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COMP24111 Machine Learning7
Probabilistic Classification
• Establishing a probabilistic model for classification (cont.)– Generative model
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GenerativeProbabilistic Model
for Class 1
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for Class 2
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for Class L
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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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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
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COMP24111 Machine Learning10
Naïve Bayes
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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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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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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
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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…