Gaussian Classifier¶
Used for classifying continuous data
\[ \begin{aligned} P(C|x) & \propto P(x|C) \times P(C) \\ & \propto N(x; \mu_c, \sigma^2_c) P(C) \\ \underbrace{P(C)}_{\text{Posterior}} & \propto \underbrace{ \frac{1}{\sqrt{2\pi \sigma^2_c}} \ \exp \left( \frac{-(x-\mu_c)^2}{2\sigma^2_c} \right) }_{\text{Likelihood}} \underbrace{P(C)|x}_{\text{Prior}} \end{aligned} \]
However equation is not used as it is; we take \(\log\) on both sides and find log likehood
\[ \begin{aligned} \text{LL}(x|C) &= \text{LL}(x|\mu_c, \sigma_c^2) \\ &= \ln P(x | \mu_c, \sigma_c^2) \\ &= \ln \left[ \frac{1}{\sqrt{2 \pi \sigma_c^2}} \right] \ \exp somethign \end{aligned} \]
\[ \text{LL} \underbrace{(C|x)}_\text{Posterior} = \text{LL} \underbrace{(x|C)}_\text{Likelihood} + \text{LR} \underbrace{(C)}_\text{Posterior} \]
2 Classes¶
\[ \begin{aligned} \ln \frac{P(C_1 | x)}{P(C_2 | x)} &= \ln P(C_1 | x) - \ln P(C_2 | x) \\ &= \frac{-1}{2} () \end{aligned} \]
- If log ratio \(\ge 0\), assign to \(C_1\)
- If log ratio \(<0\), assign to \(C_2\)
We need to ensure that we have equal sample of both classes, so that the prior probabilities of both the classes in the formula is the same.