18 Bayes rule as a classifier

Denote by $x$ an observed feature vector (data), and $C$ a random variable representing the class label.
Bayes’ rule :

$$P(C\mid x)=\frac{P(x\mid C)P(C)}{P(\mathbf{x})}$$

• $P(C|\mathbf{x})$ : Posterior probability of class $C$ given feature vector $X$.
• $P(\mathbf{x}|C)$ : Likelihood of feature vector $\mathbf{x}$ given class C.
• $P(C)$ : Prior probability of class C.
• $P(\mathbf{x})$ : Marginal likelihood, also classed the Evidence (constant with respect to C).