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Improve Naïve Bayesian Classifier by Discriminative Training

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Improve Naïve Bayesian Classifier by Discriminative Training

Kaizhu Huang, Zhangbing Zhou,

Irwin King, Michael R. Lyu

Oct. 2005

ICONIP 2005

- Background
- Classifiers
- Discriminative classifiers: Support Vector Machines
- Generative classifiers: Naïve Bayesian Classifiers

- Classifiers
- Motivation
- Discriminative Naïve Bayesian Classifier
- Experiments
- Discussions
- Conclusion

ICONIP 2005

SVM

- Discriminative Classifiers
- Directly maximize a discriminative function or posterior function
- Example: Support Vector Machines

ICONIP 2005

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

Constant w.r.t. C

Combining the assumption

ICONIP 2005

- Comparison

Example of Missing Information:

From left to right: Original digit, 50% missing digit, 75% missing digit, and occluded digit.

ICONIP 2005

Training set

subset D1

labeled as Class 1

subset D2

Labelled as Class 2

Needed!

Estimate distribution P1 to

approximate D1

Estimate distribution P2 to

approximate D2

Construct Bayes rule for

classification

- 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

ICONIP 2005

Scheme for Generative classifiers in two-category classification tasks

- 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

A-T is thus the known features

ICONIP 2005

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 Classifier in a discriminative way.

ICONIP 2005

Training set

Sub-set D1

labeled as Class I

Sub-set D2

labeled as Class 2

Interaction is needed!!

Estimate the

distribution P1 to

approximate D1

Estimate the

distribution P2 to

approximate D2

Use Bayes rule for

classification

Easily solved by Lagrange Multiplier method

ICONIP 2005

Mathematic Explanation of Naïve Bayesian Classifier

Working Scheme of Naïve Bayesian Classifier

- 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

ICONIP 2005

- Complete Optimization problem

Nonlinear optimization problem under linear constraints.

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- Solve the Optimization problem
- Using Rosen Gradient Projection methods

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Gradient and Projection matrix

ICONIP 2005

ICONIP 2005

- Experimental Setup
- Datasets
- 4 benchmark datasets from UCI machine learning repository

- Experimental Environments
- Platform:Windows 2000
- Developing tool: Matlab 6.5

- Datasets

ICONIP 2005

- Observations
- DNB outperforms NB in every datasets
- DNB wins in 2 datasets while it loses in the other 2 datasets in comparison with SVM
- SVM outperforms DNB in Segment and Satimages

ICONIP 2005

- Scheme
- 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)

- DNB uses

…………..(5)

ICONIP 2005

Setup : Randomly discard features gradually from a small percentage to a big percentage

Error Rate in Iris with missing information

Error Rate in Vote with missing information

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Error Rate in Satimage with missing information

Error Rate in DNA with missing information

ICONIP 2005

- 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.

ICONIP 2005

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

- Discriminative training on constrained Bayesian Network Classifier is possible…

ICONIP 2005

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

ICONIP 2005