Simple Bayesian Classifiers Do Not Assume Independence

Bayes’ theorem tells us how to optimally predict the class of a previously unseen example, given a training sample. The chosen class should be the one which maximizes P(CilE) = P(Ci) P(EICi) /P(E), where Ci is the ith class, E is the test example, P(YIX) denotes the conditional probability of Y given X, and probabilities are estimated from the training sample. Let an example be a vector of a attributes. If the attributes are independent given the class, P(EICi) can be decomposed into the product P(vlICi) . . . P(va(Ci), where v.j is the value of the jth attribute in the example E. Therefore we should predict the class that maximizes: