Linear Discriminant Analysis

Gaussian

Assuming all class have the same variance \(\sigma^2\), the Bayes decision boundary for two classes \((A,B)\) goes as,

\[\begin{split} \hat y(x) = \begin{cases} A, & \frac{\mu_A - \mu_B}{\sigma^2}x - \frac{|\mu_A|^2 - |\mu_B|^2}{2\sigma^2} + \log(\pi_A) - \log(\pi_B) > 0\\ B & \text{otherwise} \end{cases} \end{split}\]

The posterior probability for any class follows the logistic function.