1 data classification and segmentation : bayesian methods iii hk book: section 7.4, domingos’...

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Data Classification and Segmentation: Bayesian Methods III

HK Book: Section 7.4, Domingos’ Paper (online)

Instructor: Qiang YangHong Kong University of Science and Technology

[email protected]

Thanks: Dan Weld, Eibe Frank

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First, Review Naïve Bayesian

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Naïve Bayesian is Surprisingly Good. Why?

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Independence Test Naïve Bayesian’s good performance may be

due to independence of attributes? Modeling independence of attributes Am and

An

D() is zero when completely independent D() is large if dependent

)Pr(log)Pr()Pr()|(

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How to measure independence?

H(A|C): the once the class C is given, how much is A given?

In other words, how random is A once C is given We know randomness can be measured in Entropy Entropy is –p*log(p), and then summed over all

possible values Thus, to measure the dependence of A on C,

we can look at each value Ci of C, such that under Ci,

Find the entropy of A Then, sum over all possible Ci

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Measuring dependency

Example: C=Play attribute, A=Windy Attribute

C1=yes, C2=no Pr(C=Yes)=9/14 When C=C1 (Play = yes)

A=True: 3 counts A=False: 6 counts Pr(A=True and C=C1)=3/14 Pr(A=False and C=C1)=6/14

When C=C2 (no) Pr(A=True and C=C2)=3/14 Pr(A=False and C=C2)=2/14

H(A|C)=9/14*[(-3/14)log(3/14)-(6/14)log(6/14)]+5/14*[-3/14log(3/14)-2/14log(2/14)]

Windy Play

FALSE no

TRUE no

FALSE *yes

FALSE *yes

FALSE *yes

TRUE no

TRUE *yes

FALSE no

FALSE *yes

FALSE *yes

TRUE *yes

TRUE *yes

FALSE *yes

TRUE no

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Property of H(A|C)

If A is completely dependent on C, what is H(A|C)? In the extreme case, whenever C=yes, A is true;

whenever C=no, A=false. Pr(C=yes and A=true)=1 Pr(C=yes and A=false)=0 H(A|C)=?

The other extreme: if A is completely independent on C, what is H(A|C)?

Pr(…) between 0 and 1, H(A|C)=??

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Extending to two attributes For A1 and A2, we simply consider the

Cartesian product of their values This gives a single attribute Then we measure the dependency of A1A2 (a

new attribute) on class C Example: Consider A1=Humidity and

A2=Windy, Class= Play Humidity={high, normal} Windy={True, False} A1A2=HumidityWindy={hightrue, highfalse,

normaltrue, normalfalse}

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Humidity

HumidityWindy

Play

high highFALSE no

high highTRUE no

high highFALSE yes

high highFALSE yes

normal normal FALSE yes

normal normal TRUE no

normal normal TRUE yes

high high FALSE no



normal normal TRUE yes

high high TRUE yes


high high TRUE no

HumidityWindy Play

highFALSE no

highTRUE no

highFALSE yes

highFALSE yes

normal FALSE yes

normal TRUE no

normal TRUE yes

high FALSE no

normal FALSE yes

normal FALSE yes

normal TRUE yes

high TRUE yes

normal FALSE yes

high TRUE no

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Putting them together

When the two attributes Am and An are independent, given C,

H(AmAn |C)=H(Am|C)+H(An|C) D(…) = 0

When they are dependent D(…) is a large value

Thus, D (…) is a measure of dependency between attributes

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Most domains are not independent

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Why Perform So well? (section 4 of paper)

Assume three attributes A, B, C Two classes: + and – (say, play=+ means yes) Assume A and B are the same – completely dependent Assume Pr(+)=Pr(-)=0.5 Assume that A and C are independent

Thus, Pr(A,C|+)=Pr(A|+)*Pr(C|+) Optimal Decision:

If Pr(+)*Pr(A,B,C|+)>Pr(-)*Pr(A,B,C|-), then answer =+; else answer=-

Pr(A,B,C|+)=Pr(A,A,C|+)=Pr(A,C|+)=Pr(A|+)*Pr(C|+) Likewise for – Thus, Optimal method is:

Pr(A|+)*Pr(C|+) > Pr(A|-)*Pr(C|-)

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Analysis If we use the Naïve Bayesian method,

IF Pr(+)*Pr(A|+)*Pr(B|+)*Pr(C|+)> Pr(-)*Pr(A|-)*Pr(B|-)*Pr(C|-)

Then answer = + Else, answer = -

Since A=B, and Pr(+)=Pr(-), we have Pr(A)2*Pr(C|+)> Pr(A)2*Pr(C|-)

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Simplify the NB Formula

Naïve Bayesian Formula Pr(A)2*Pr(C|+)> Pr(A)2*Pr(C|-) becomes

p2q > (1-p)2q (Eq 2) Thus, our question is:

In order to know why Naïve Bayesian perform so well, we want to ask:

When does the optimal decision agree (or differ) with Naïve Bayesian decision?

That is, where do formulas (Eq 1) and (Eq 2) agree or disagree?

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disagree

disagree

Optimal

Optimal

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Conclusion

In most cases, naïve Bayesian performs the same as the optimal classifiers

That is, the error rates are minimal This has been confirmed in many practical

applications

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Applications of Bayesian Method

Gene Analysis Nir Friedman Iftach Nachman Dana Pe’er, Institute of

Computer Science, Hebrew University Text and Email analysis

Spam Email Filter Microsoft Work

News classification for personal news delivery on the Web User Profiles

Credit Analysis in Financial Industry Analyze the probability of payment for a loan

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Gene Interaction Analysis DNA

Gene

DNA is a double-stranded molecule Hereditary information is encoded Complementation rules

Gene is a segment of DNA Contain the information required to make a protein

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Gene Interaction Result: Example of interaction

between proteins for gene SVS1.

The width of edges corresponds to the conditional probability.

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Spam Killer Bayesian Methods are used for weed out spam

emails

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Spam Killer

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Construct your training data

Each email is one record: M Emails are classified by user into

Spams: + class Non-spams: - class

A email M is a spam email if Pr(+|M)>Pr(-|M)

Features: Words, values = {1, 0} or {frequency} Phrases Attachment {yes, no}

How accurate: TP rate > 90% We wish FP rate to be as low as possible Those are the emails that are nonspam but are classified

as spam

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Naïve Bayesian In Oracle9ihttp://otn.oracle.com/products/oracle9i/htdocs/o9idm_faq.html

What is the target market?Oracle9i Data Mining is best suited for companies that have lots of data, are committed to the Oracle platform, and want to automate and operationalize their extraction of business intelligence. The initial end user is a Java application developer, although the end user of the application enhanced by data mining could be a customer service rep, marketing manager, customer, business manager, or just about any other imaginable user.

What algorithms does Oracle9i Data Mining support?Oracle9i Data Mining provides programmatic access to two data mining algorithms embedded in Oracle9i Database through a Java-based API. Data mining algorithms are machine-learning techniques for analyzing data for specific categories of problems. Different algorithms are good at different types of analysis. Oracle9i Data Mining provides two algorithms: Naive Bayes for Classifications and Predictions and Association Rules for finding patterns of co-occurring events. Together, they cover a broad range of business problems.

Naive Bayes: Oracle9i Data Mining's Naive Bayes algorithm can predict binary or multi-class outcomes. In binary problems, each record either will or will not exhibit the modeled behavior. For example, a model could be built to predict whether a customer will churn or remain loyal. Naive Bayes can also make predictions for multi-class problems where there are several possible outcomes. For example, a model could be built to predict which class of service will be preferred by each prospect.

Binary model example:Q: Is this customer likely to become a high-profit customer?A: Yes, with 85% probability

Multi-class model example:Q: Which one of five customer segments is this customer most likely to fit into — Grow, Stable, Defect, Decline or Insignificant?A: Stable, with 55% probability

1 data classification and segmentation : bayesian methods iii hk book: section 7.4, domingos’...

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