Bayes Optimal Classification

Bayes optimal classification \[ \arg \max_{v_{j} \in V} \sum_{h_{i} \in H} P(v_{j}\,|\,h_{i}) P(h_{i}\,|\,D)\]
Example: \[ \array{ P(h_{1}\,|\,D)=.4, & P(\ominus \,|\,h_{1})=0, & P(\oplus \,|\,h_{1})=1 \\ P(h_{2}\,|\,D)=.3, & P(\ominus \,|\,h_{2})=1, & P(\oplus \,|\,h_{2})=0 \\ P(h_{3}\,|\,D)=.3, & P(\ominus \,|\,h_{3})=1, & P(\oplus \,|\,h_{3})=0 } \] therefore \[ \array{ \sum_{h_{i} \in H} P(\oplus \,|\,h_{i}) P(h_{i}\,|\,D) & = & .4 \\ \sum_{h_{i} \in H} P(\ominus \,|\,h_{i}) P(h_{i}\,|\,D) & = & .6 } \] and \[ \array{ \arg \max_{v_{j} \in V} \sum_{h_{i} \in H} P(v_{j}\,|\,h_{i}) P(h_{i}\,|\,D) & = & \ominus }\]