Discriminative Naïve Bayesian Classifiers

1. Discriminative Naïve Bayesian Classifiers Kaizhu Huang Supervisors: Prof. Irwin King, Prof. Michael R. Lyu Markers: Prof. Lai Wan Chan, Prof. Kin Hong Wong

2. Outline • Background • Classifiers • Discriminative classifiers: Support Vector Machines • Generative classifiers: Naïve Bayesian Classifiers • Motivation • Discriminative Naïve Bayesian Classifiers • Experiments • Discussions • Conclusion

3. SVM Background • Discriminative Classifiers • Directly maximize a discriminative function or posterior function • Example: Support Vector Machines

4. Constant w.r.t. C Combining the assumption Background • Generative Classifiers • Model the joint distribution for each class P(x|C) and then use Bayes rules to construct posterior classifiers P(C|x). • Example: Naïve Bayesian Classifiers • Model the distribution for each class under the assumption: each feature of the data is independent with others features, when given the class label.

5. Background • Comparison Example of Missing Information: From left to right: Original digit, 50% missing digit, 75% missing digit, and occluded digit.

6. Pre-classified dataset Sub-dataset D1 for Class 1 Sub-dataset D2 for Class 2 Estimate the distribution P1 to approximate D1 accurately Estimate the distribution P2 to approximate D2 accurately Use Bayes rule to perform classification Background • Why Generative classifiers are not accurate as Discriminative classifiers? It is incomplete for generative classifiers to just approximate the inner-class information. The inter-class discriminative information between classes are discarded Scheme for Generative classifiers in two-category classification tasks

7. Background • Why Generative Classifiers are superior to Discriminative Classifiers in handling missing information problems? • SVM lacks the ability under the uncertainty • NB can conduct uncertainty inference under the estimated distribution. A is the feature set T is the subset of A, which is missing

8. Motivation • It seems that a good classifier should combine the strategies of discriminative classifiers and generative classifiers. • Our work trains one of the generative classifier: Naïve Bayesian Classifies in a discriminative way.

9. Roadmap of our work Discriminative training

10. Discriminative Classifiers HMM and GMM Generative Classifiers Discriminative training 1. 2. How our work relates to other work? Jaakkola and Haussler NIPS98 Difference: Our method performs a reverse process: From Generative classifiers to Discriminative classifiers Beaufays etc., ICASS99, Hastie etc., JRSS 96 Difference: Our method is designed for Bayesian classifiers.

11. How our work relates to other work? Optimization on Posterior Distribution P(C|x) 3. Logistical Regression (LR) Difference: LR will encounter computational difficulties in handling missing information problems. When number of the missing or unknown features grows, it will be intractable to perform inference.

12. Roadmap of our work

13. Pre-classified dataset Sub-dataset D1 for Class I Sub-dataset D2 for Class 2 Estimate the distribution P1 to approximate D1 accurately Estimate the distribution P2 to approximate D2 accurately Use Bayes rule to perform classification Discriminative Naïve Bayesian Classifiers Easily solved by Lagrange Multiplier method Mathematic Explanation of Naïve Bayesian Classifier Working Scheme of Naïve Bayesian Classifier

14. Discriminative Naïve Bayesian Classifiers (DNB) • Optimization function of DNB Divergence item • On one hand, the minimization of this function tries to approximate the dataset as accurately as possible. • On the other hand, the optimization on this function also tries to enlarge the divergence between classes. • Optimization on joint distribution directly inherits the ability of NB in handling missing information problems

15. Discriminative Naïve Bayesian Classifiers (DNB) • Complete Optimization problem Cannot separately optimize and as in NB, Since they are interactive variables now.

16. Discriminative Naïve Bayesian Classifiers (DNB) • Solve the Optimization problem • Nonlinear optimization problem under linear constraints. Using Rosen Gradient Projection methods

17. Discriminative Naïve Bayesian Classifiers (DNB) Gradient and Projection matrix

18. Extension to Multi-category Classification problems

19. Experimental results • Experimental Setup • Datasets • 5 benchmark datasets from UCI machine learning repository • Experimental Environments • Platform:Windows 2000 • Developing tool: Matlab 6.5

20. Without information missing • Observations • DNB outperforms NB in every datasets • DNB wins in 2 datasets while it loses in three dataets in comparison with SVM • SVM outperforms DNB in Segment and Satimages

21. With information missing • DNB uses to conduct inference when there is information missing • SVM sets 0 values to the missing features (the default way to process unknown features in LIBSVM)

22. With information missing

23. With information missing

24. With information missing

25. With information missing • Observations • NB demonstrates a robust ability in handling missing information problems. • DNB inherits the ability of NB in handling missing information problems while it has a higher classification accuracy than NB • SVM cannot deal with missing information problems easily. • In small datasets, DNB demonstrates a superior ability than NB.

26. SVM Discussion • Why SVM outperforms DNB when no information missing? SVM DNB • SVM directly minimizes the error rate, while DNB minimizes an intermediate term. • SVM assumes no model, while DNB assumes independent relationship among features. “all models are wrong but some are useful”.

27. Discussion • How DNB relates to Fisher Discriminant (FD)? FD • Using the difference of the mean between two classes as the divergence measure is not an informative way in comparison with using distributions. • FD is usually used as dimension reduction method rather than a classification method

28. Discussion • Can DNB be extended to general Bayesian Network (BN) Classifier? • Finding optimal General Bayesian Network Classifiers is an NP-complete problem. • Structure learning problem will be involved. Direct application of DNB will encounter difficulties since the structure is non-fixed in restricted BNs . • The tree-like discriminative Bayesian Network Classifier is ongoing.

29. Discussion Discriminative training of Tree-like Bayesian Network Classifiers And as far as possible from the distribution of the other dataset Two reference distributions are used in each iteration. Approximate the Empirical distribution as close as possible

30. Future work • Extensive evaluations on discriminative Bayesian network classifiers including Discriminative Naïve Bayesian Classifiers and tree-like Bayesian Network Classifiers.

31. Conclusion • We develop a novel model named Discriminative Naïve Bayesian Classifiers • It outperforms Naïve Bayesian Classifiers when no information is missing • It outperforms SVMs in handling missing information problems.