# Bayes classifier in terms of generative and discriminant analysis approaches

For simplicity I will only discuss binary classification. If $p_k(x) = P(X \mid Y = k)$ for $k = 0,1$, then Bayes classifier $h$ minimizes the risk $P(h(x) \neq Y)$. It is well known that $$h(x) = 1\{ \frac{p_1(x)}{p_0(x)} > \frac{1 - \pi_1}{\pi_1} \},$$ where $\pi_1 = p_0(x)$. Bayes classifier seems to naturally fit the generative approach where we directly model $p_k(x)$ and $\pi_k$. Where does the discriminant analysis approach fall in this framework? Is the discriminant analysis approach completely independent of Bayes' classifier? What exactly is the connection between these three topics?