Editing Naive Bayes classifier (section)

=== Constructing a classifier from the probability model ===

The discussion so far has derived the independent feature model, that is, the naive Bayes [[probability model]]. The naive Bayes [[Statistical classification|classifier]] combines this model with a [[decision rule]]. One common rule is to pick the hypothesis that is most probable so as to minimize the probability of misclassification; this is known as the ''[[maximum a posteriori|maximum ''a posteriori'']]'' or ''MAP'' decision rule. The corresponding classifier, a [[Bayes classifier]], is the function that assigns a class label <math>\hat{y} = C_k</math> for some {{mvar|k}} as follows:
<math display="block">\hat{y} = \underset{k \in \{1, \ldots, K\}}{\operatorname{argmax}} \ p(C_k) \displaystyle\prod_{i=1}^n p(x_i \mid C_k).</math>
[[File:ROC_curves.svg|thumb|[[Likelihood function]]s <math>p(\mathbf{x} \mid Y)</math>, [[Confusion matrix]] and [[ROC curve]]. For the naive Bayes classifier and given that the a priori probabilities <math>p(Y)</math> are the same for all classes, then the [[decision boundary]] (green line) would be placed on the point where the two probability densities intersect, due to {{nowrap|<math>p(Y \mid \mathbf{x}) = \frac{p(Y) \ p(\mathbf{x} \mid Y)}{p(\mathbf{x})} \propto p(\mathbf{x} \mid Y)</math>.}}]]