1 Data Classification and Segmentation : Bayesian Methods III HK Book: Section 7.4, Domingos’ Paper (online) Instructor: Qiang Yang Hong Kong University of Science and Technology Thanks: Dan Weld, Eibe Frank

2

3 First, Review Naïve Bayesian

4 Naïve Bayesian is Surprisingly Good. Why?

5 Independence Test Naïve Bayesian’s good performance may be due to independence of attributes? Modeling independence of attributes A m and A n D() is zero when completely independent D() is large if dependent

6 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

7 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)] WindyPlay FALSEno TRUEno FALSE*yes FALSE*yes FALSE*yes TRUEno TRUE*yes FALSEno FALSE*yes FALSE*yes TRUE*yes TRUE*yes FALSE*yes TRUEno

8 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)=??

9 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}

10 HumidityHumidityWindyPlay highhighFALSEno highhighTRUEno highhighFALSEyes highhighFALSEyes normalnormal FALSEyes normalnormal TRUEno normalnormal TRUEyes highhigh FALSEno normalnormal FALSEyes normalnormal FALSEyes normalnormal TRUEyes highhigh TRUEyes normalnormal FALSEyes highhigh TRUEno HumidityWindyPlay highFALSEno highTRUEno highFALSEyes highFALSEyes normal FALSEyes normal TRUEno normal TRUEyes high FALSEno normal FALSEyes normal FALSEyes normal TRUEyes high TRUEyes normal FALSEyes high TRUEno

11 Putting them together When the two attributes A m and A n are independent, given C, H(A m A n |C)=H(A m |C)+H(A n |C) D(…) = 0 When they are dependent D(…) is a large value Thus, D (…) is a measure of dependency between attributes

12 Most domains are not independent

13 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|-)

14 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|-)

15 Simplify Optimal Formula Let Pr(+|A)=p, Pr(+|C)=q Pr(A|+)=Pr(+|A)*Pr(A)/Pr(+)=p*Pr(A)/Pr(+) Pr(A|-)=(1-p)*Pr(A)/Pr(-) Pr(C|+)=Pr(+|C)*Pr(C)/Pr(+)=q*Pr(C)/Pr(+) Pr(C|-)=(1-q)*Pr(C)/Pr(-) Thus, the optimal method If Pr(+)*Pr(A|+)*Pr(C|+)>Pr(-)*Pr(A|-)*Pr(C|-), then answer =+; else answer=- becomes p*q > (1-p)*(1-q) (Eq1)

16 Simplify the NB Formula Naïve Bayesian Formula Pr(A) 2 *Pr(C|+)> Pr(A) 2 *Pr(C|-) becomes p 2 q > (1-p) 2 q (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?

17 disagree Optimal

18 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

19 Applications of Bayesian Method Gene Analysis Nir Friedman Iftach Nachman Dana Pe’er, Institute of Computer Science, Hebrew University Text and analysis Spam 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

20 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

21 Gene Interaction Result: Example of interaction between proteins for gene SVS1. The width of edges corresponds to the conditional probability.

22 Spam Killer Bayesian Methods are used for weed out spam s

23 Spam Killer

24 Construct your training data Each is one record: M s are classified by user into Spams: + class Non-spams: - class A M is a spam 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 s that are nonspam but are classified as spam

25 Naïve Bayesian In Oracle9i 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