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Classifier Clarifier

Classifier Clarifier. Bayesian Classifiers. The Big Picture. Problem : classify data into one of four classes Data : you select 3 features which are mutually exclusive. {Note : this is a really bad decision. Why? We’ll fix it later} You generate feature vectors for the training data.

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Classifier Clarifier

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  1. Classifier Clarifier Bayesian Classifiers

  2. The Big Picture • Problem : classify data into one of four classes • Data : you select 3 features which are mutually exclusive. {Note : this is a really bad decision. Why? We’ll fix it later} • You generate feature vectors for the training data.

  3. The Big Picture Cont. • Once you have the classifier, you will observe features in the testing vectors and based on those assign a class to the vector. • In other words, we want to find the class (Ci) given the feature observed (Xk). • We need to find a formula for P(Ci | Xk)

  4. The Feature Vectors

  5. Training Data Summary

  6. Fundamental Probabilities • P(C1) = .4 P(X1) = .44 • P(C2) = .16 P(X2) = .4 • P(C3) = .24 P(X3) = .16 • P(C4) = .20 • Sum of P(Ci) = 1 because all vectors are in one of the classes • Sum P(Xi) = 1 because each vector has only one feature

  7. Question • Are the features and the classes independent? • Definition of independent is : • P(a AND b) = P(a)*P(b) • Check : P(C1 AND X1) = 6/25 = .24 from table • But P(C1) * P(X1) = (.44)*(.4) = .176 • The features and classes are not independent

  8. So? Need Conditional Probability • P(C AND X) = P(C|X) * P(X) or • P(X AND C) = P(X|C) * P(C) • P(C AND X ) = P(X AND C) • P(C|X) * P(X) = P(X|C) *P(C) • P(C|X) = [P(X|C) * P(C)] /P(X) • Now we have what we need

  9. The Classifier • For each feature, determine which class is most likely given that the feature was present. • Calculate the probability of being in each class based on that feature’s presence • Select the largest probability • Classify the sample • Note : we need the features to be mutually exclusive to make this works. We will fix this in a minute.

  10. The probabilities • P(C1|X1) = P(X1|C1)*P(C1) = (.6)*(.4) = .24 • P(C1|X2) = P(X2|C1)*P(C1) = (.3)*(.4) = .12 • P(C1|X3) = P(X3|C1)*P(C1) = (.1)*(.4) = .04 • P(C2|X1) = P(X1|C2)*P(C2) = (0)*(.16) = 0 • P(C2|X2) = P(X2|C2)*P(C2) = (.5)*(.16) = .08 • P(C2|X3) = P(X3|C2)*P(C2) = (.5)*(.16) = .08 • P(C3|X1) = P(X1|C3)*P(C3) = (.17)*(.24) = .04 • P(C3|X2) = P(X2|C3)*P(C3) = (.67)*(.24) = .16 • P(C3|X3) = P(X3|C3)*P(C3) = (.17)*(.24) = .04 • P(C4|X1) = P(X1|C4)*P(C4) = (.8)*(.2) = .16 • P(C4|X2) = P(X2|C4)*P(C4) = (.2)*(.2) = .04 • P(C4|X3) = P(X3|C4)*P(C4) = (0)*(.2) = 0

  11. Examples • Test vector 1 = 100 • P(C1|X1) = .24 • P(C2|X1) = 0 • P(C3|X1) = .04 • P(C4|X1) = .16 • Classify as C1 • Test vector 2 = 010 • P(C1|X2) = .12 • P(C2|X2) = .08 • P(C3|X2) = .16 • P(C4|X2) = .04 • Classify as C3 • Test vector 3 = 001 • P(C1|X3) = .04 • P(C2|X3) = .08 • P(C3|X3) = .04 • P(C4|X3) = 0 • Classify as C2 • You guessed it : we never classify anything as class 4.

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