Bayesian Discriminant Function

The Bayesian (minimum error rate) discriminant function $g(\mathbf{x})$ is a decision rule that selects the class with the highest posterior probability given a feature vector $\mathbf{x}$ :

$$g(\mathbf{x})=\arg \underset{i}{\max }P(s=i\mid \mathbf{x})$$

In contrast, $g_i(\mathbf{x})$ is the discriminant function for a specific class $i$, which calculates a score for that class:

$$g_i(\mathbf{x})=P(s=i\mid \mathbf{x})$$

Classification is performed by computing $g_i(\mathbf{x})$ for each class and selecting the class with the highest value. This approach minimizes the overall probability of classification error.

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Class Problems

$$\boxed{s^{\ast }=\arg \underset{i\in \mathcal{S}}{\max }{g}_i(\mathbf{x})}$$

where, $g_i(\mathbf{x})=P(s=i|\mathbf{x})$, gives the minimum error rate classification.

Decision Problems

$$\boxed{a^{\ast }=\arg \underset{j\in \mathcal{A}}{\max }{g}_i(\mathbf{x})}$$

where, $g_i(\mathbf{x})=-R(a=i|\mathbf{x})$, gives the minimum risk decision.