# How to use Conjugate Gradient Method to maximize log marginal likelihood

I'm trying to maximize the log marginal likelihood of a Gaussian process with respect to its hyper parameters (with a squared exponential kernel, to be specific). I've been referring to the text Gaussian Processes for Machine Learning by Rasmussen & Williams to try to get me through this problem, and I see they refer to the Conjugate Gradient Method very often to maximize the log marginal likelihood, but they don't explain how exactly. Maybe it's obvious, but I'm not seeing it.

The Conjugate Gradient Method solves problems of the form $Ax=b$, which turns out is the same as maximizing $f(x) = 1/2 x ^{T}Ax - x^{T}b$. However, I don't see how either of these equations is particularly relevant to the problem I'm trying to solve. The log marginal likelihood only has one term that's quadratic, so how would one maximize the log marginal likelihood while taking the other terms into account as well? And the covariance matrix is constantly changing with the hyper parameters, not the original data itself -- so I'm not trying to solve for $x$ necessarily in $f(x) = 1/2 x ^{T}Ax - x^{T}b$ which seems to be the goal of the Conjugate Gradient Method?

$\min f(x)$