Bayeses optimal classifier

Suppose we have a classification problem for some function $f: A \rightarrow B$. We have a hypothesis space $H$ and training data $T$. We want to work out what the best label is for $a \in A$. We can use maximum a posteriori probability to calculate Bayeses optimal classifier

$$v_{MAP} = \mbox{arg}\max_{v \in V} \sum_{h \in H} \mathbb{P}[v \vert h] \mathbb{P}[h \vert T].$$