Quadratic Discriminant Analysis

Definition

$\underset{1\le k\le K}{\mathrm{argmax}}{\hat{f}}_k(x)=\underset{1\le k\le K}{\mathrm{argmax}}\left[\ln {\hat{\pi }}_k-\frac{1}{2}\ln |{\hat{\Sigma }}_k|-\frac{1}{2}(x-{\hat{\mu }}_k{)}^{⊺}{\hat{\Sigma }}_k^{-1}(x-{\hat{\mu }}_k)\right]$ where ${\hat{\pi }}_k=\frac{n_k}{n}$ is the sample ratio, ${\hat{\mu }}_k=\frac{1}{n_k}\sum _{i:Y_i=k}{X}_i$ is the sample mean, and ${\hat{\Sigma }}_k=\frac{1}{n_k-1}\sum _{i:Y_i=k}(X_i-{\hat{\mu }}_k)(X_i-{\hat{\mu }}_k{)}^{⊺}$ is the sample variance covariance matrix

Bayes Classifier with an assumption that $p_k(x)\sim N_p({\mu }_k,{\Sigma }_k)$