$$ P(y \mid \mathbf{X}) \propto P(\mathbf{X} \mid y)\,P(y) \\[6pt]$$
$$ P(\mathbf{X} \mid y) = \prod_{j=1}^{d} P(x_j \mid y) \\[6pt] $$
$$ P(x_j \mid y = k) = \frac{1}{\sqrt{2\pi\sigma_{kj}^2}} \exp\!\left(-\frac{(x_j - \mu_{kj})^2}{2\sigma_{kj}^2}\right) \\[8pt] $$
$$\mu_{kj} = \frac{1}{N_k}\sum_{i:y_i=k} x_{ij}$$
$$\sigma_{kj}^2 = \frac{1}{N_k}\sum_{i:y_i=k}(x_{ij}-\mu_{kj})^2, \quad$$
$$P(y=k)=\frac{N_k}{N}$$
| x1 | x2 | Class | x1 | x2 | Class |
|---|---|---|---|---|---|
| 5 | 3 | c1 | 6 | 7 | c2 |
| 5 | 4 | c1 | 8 | 7 | c2 |
| 5 | 5 | c1 | 9 | 6 | c2 |
| 5 | 7 | c1 | 10 | 7 | c2 |
| 5 | 9 | c1 | 11 | 8 | c2 |
Predict
| x1 | x2 | Class |
|---|---|---|
| 10 | 4 | ? |
| 5 | 12 | ? |