I'm reading the An Introduction to Statistical Learning book and get stuck at Chapter 4 page 159 where the author(s) state that in the QDA setting, the log odds of the posterior probabilities is quadratic in x:
$$ log \left( \dfrac{Pr(Y = k|X = x)}{Pr(Y = K|X = x)} \right) = a_k + \sum _{j=1}^{p} b_{kj} x_j + \sum _{j=1}^{p} \sum _{l=1}^{p} c_{kjl} x_j x_l$$
where $$ a_k, b_{kj}, c_{kjl} $$ are functions of $$ \pi_k , \pi_K, \mu_k, \mu_K, \Sigma_k, \Sigma_K $$
In exercise 11 (page 192), I am asked to work out the form of $$ a_k, b_{kj}, c_{kjl} $$ and I have no idea how to start. How can I solve this problem?
Thank you.
Link for the book: https://hastie.su.domains/ISLR2/ISLRv2_website.pdf
Ratio of posteriors gives $$ \frac{\pi_k \sqrt{|\mathbf{\Sigma}_K|}} {\pi_K \sqrt{|\mathbf{\Sigma}_k|}} \exp \left[ - \frac12 (\mathbf{x}-\mathbf{m}_k)^T \mathbf{\Sigma}_k^{-1}(\mathbf{x}-\mathbf{m}_k) +\frac12 (\mathbf{x}-\mathbf{m}_K)^T \mathbf{\Sigma}_K^{-1}(\mathbf{x}-\mathbf{m}_K) \right] $$ Taking log yields $$ a + \frac12 \left[ (\mathbf{x}-\mathbf{m}_K)^T \mathbf{\Sigma}_K^{-1}(\mathbf{x}-\mathbf{m}_K) - (\mathbf{x}-\mathbf{m}_k)^T \mathbf{\Sigma}_k^{-1}(\mathbf{x}-\mathbf{m}_k) \right] = a + \frac12 \left[ \mathbf{x}^T \mathbf{Q} \mathbf{x} + \mathbf{b}^T \mathbf{x} + c \right] $$ You are asked to make explicit the scalars $a,c$, the vector $\mathbf{b}$ and the symmetric matrix $\mathbf{Q}$. I leave it to you to do the calculations from here.