Kernel Based Classifier

Definition

$\underset{1\le k\le K}{\mathrm{argmax}}{\hat{f}}_k(x)=\underset{1\le k\le K}{\mathrm{argmax}}{\hat{p}}_k(x){\hat{\pi }}_k=\underset{1\le k\le K}{\mathrm{argmax}}\left(\frac{1}{n}|H{|}^{-1}\sum _{i:Y_i=k}K(H^{-1}(\mathbf{x}-{\mathbf{X}}_i))\right)$ where ${\hat{p}}_k(x)=\frac{1}{n_k}|H{|}^{-1}\sum _{i:Y_i=k}K(H^{-1}(\mathbf{x}-{\mathbf{X}}_i))$ and ${\hat{\pi }}_k=\frac{n_k}{n}$

Bayes Classifier with an assumption that $p_k(x)$ is the Kernel Density Estimation.