Quadratic Discriminant Analysis

Gaussian

Let the multivariate gaussian be,

\[ P(x \mid \mu, \sigma) = \left(2 \pi \sigma^2 \right)^{d/2} \exp\left(-\frac{|x-\mu|^2}{2\sigma^2}\right) \]

Let the \(N\) classes be represented as \(c \in \{1,2,\ldots,N\}​\) such that each class has the following statistics:

• \(\mu_c\) : Mean of the class

• \(\sigma_c\) : Standard deviation of the class

• \(\pi_c\) : Prior of the class

The optimization problem can be simplified using logarithm to preserve maximization of the Bayes decision,

\[\begin{split} \begin{align*} \hat y(x) &= \mathop{\arg\max}_c \log\left( (2\pi)^{d/2} P(x \mid \mu, \sigma) \pi_c \right) \\ &= \mathop{\arg\max}_c \frac{|x-\mu_c|^2}{2\sigma_c^2} - d\log{\sigma_c} + \log{\pi_c} \end{align*} \end{split}\]

General Quadratic Function

For some quadratic function \(Q(x)\), the Bayes decision rule is given by,

\[ \hat y(x) = \mathop{\arg\max}_c Q_c(x) \]

The Bayes decision rule for the quadratic discriminant always product a posterior distribution that is a logistic function. We can easily see this by considering only two classes \((A,B)\).

\[\begin{split} \hat y(x) = \begin{cases} A & Q_A(x) - Q_B(x) > 0 \\ B & \text{otherwise} \end{cases} \end{split}\]

\[\begin{split} \begin{align*} P(Y \mid X) &= \frac{e^{Q_A(x)}}{e^{Q_A(x)} + e^{Q_B(x)}}\\ &= \frac{1}{1 + e^{Q_A(x) - Q_B(x)}} \end{align*} \end{split}\]