Linear Discriminant Analysis
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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.