# Random naive Bayes

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Random naive Bayes

Random naive Bayes extends the Naive Bayes classifier by adopting the random forest principles: random input selection (bagging, i.e. bootstrap aggregating) and random feature selection ( [Breiman, 2001] ).

Naive Bayes classifier

Naive Bayes is a probabilistic classifier simplifying Bayes' Theorem by "naively" assuming class conditional independence. Although this assumption leads to biased posterior probabilities, the ordered probabilities of Naive Bayes result in a classification performance comparable to that of classification trees and neural networks ( [Langley, Iba and Thomas, 1992] ). Nothwithstanding Naive Bayes' popularity due to its simplicity combined with high accuracy and speed, its conditional independence assumption rarely holds. There are mainly two approaches to alleviate this naivity:
1) Selecting attribute subsets in which attributes are conditionally independent (cf. Selective Bayesian Classifier [Langley and Sage, 1994] ).
2) Extending the structure of Naive Bayes to respresent attribute dependencies (cf. AODE [Webb et al. 2005] ).

Random naive Bayes' alleviation of the class conditional independence assumption

Random Naive Bayes adopts the first approach by randomly selecting a subset of attributes in which attributes are assumed to be conditionally independent. Naive Bayes' performance might benefit from this random feature selection. Analogous to AODE, Random Naive Bayes builds an ensemble, but unlinke AODE, the ensemble combines zero-dependence classifiers.

Random naive Bayes and random forest

Generalizing Random Forest to Naive Bayes, Random Naive Bayes (Random NB), is a bagged classifier combining a forest of "B" Naive Bayes. Each "b"th Naive Bayes is estimated on a bootstrap sample Sb with "m" randomly selected features. To classify an observation put the input vector down the "B" Naive Bayes in the forest. Each Naive Bayes generates posterior class probabilities. Unlike Random Forest, the predicted class of the ensemble is assessed by adjusted majority voting rather than majority voting, as each "b"th Naive Bayes delivers continuous posterior probabilities. Similar to Random Forest, the importance of each feature is estimated on the out-of-bag (oob) data.

Notes

References

* Breiman,L. (2001). Random Forests, Machine Learning, 45(1), 5&ndash;32.
* Langley, P., Iba, W. and Thomas, K. (1992). An analysis of Bayesian Classifiers, Proceedings of the Tenth National Conference on Artificial Intelligence, AAAI Press,223&ndash;228.
* Langley, P. and Sage, S. (1994). Induction of selective Bayesian Classifiers, Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence, Seattle, WA: Morgan Kaufmann.
* [http://dx.doi.org/10.1007/978-3-540-74469-6_35 Prinzie, A., Van den Poel, D. (2007). Random Multiclass Classification: Generalizing Random Forests to Random MNL and Random NB, Dexa 2007, Lecture Notes in Computer Science, 4653, 349&ndash;358.]
* [http://www.crm.ugent.be/Video_Lecture_UGent_Random_Multinomial_Logit_Generalizing_RF_to_other_methods_Prinzie_and_Van_den_Poel.htm Video lecture on Random Naive Bayes and Random MultiNomial Logit]