Naive Bayes Classifier

Assumes all instances $x \in X$ are described by attributes $\langle a_{1}, a_{2} \ldots a_{n} \rangle$.
Assume target function $f: X \rightarrow V$, where $V$ is some finite set.
Most probable value of $f(x)$ given $x = \langle a_{1}, a_{2} \ldots a_{n} \rangle$ is: \[ \array{ v_{MAP} &= &\argmax_{v_{j} \in V} P(v_{j} \,|\, a_{1}, a_{2} \ldots a_{n}) \\ v_{MAP} &= &\argmax_{v_{j} \in V} \frac{P(a_{1}, a_{2} \ldots a_{n}\,|\,v_{j}) P(v_{j})}{P(a_{1}, a_{2} \ldots a_{n})} \\ v_{MAP} &= &\argmax_{v_{j} \in V} P(a_{1}, a_{2} \ldots a_{n}\,|\,v_{j}) P(v_{j}) } \]
Naive Bayes assumes conditional independence given the target value, that is \[ P(a_{1}, a_{2} \ldots a_{n}\,|\,v_{j}) = \prod_{i} P(a_{i} \,|\, v_{j}) \]
Naive Bayes classifier: \[ v_{NB} = \argmax_{v_{j} \in V} P(v_{j}) \prod_{i} P(a_{i} \,|\, v_{j}) \]