(\bm{\mu}, \bm{\nu})$. (Note that $\bm{K}$ and $\bm{r}$ depend on $\bm{\theta}$.) By definition, the likelihood is $$\begin{aligned}
    L_{\bm{\Theta}}(\bm{\theta}; \bm{y}) = f_{\bm{Y}_{N}}(\bm{y}; \bm{\theta})\end{aligned}$$ and hence $$\begin{aligned}
    &\ln L_{\bm{\Theta}}(\bm{\theta}; \bm{y}) = - \dfrac{1}{2} (\bm{y} - \bm{r})^{\mathrm{t}} \bm{K}^{-1} (\bm{y} - \bm{r}) - \dfrac{1}{2} \ln |\bm{K}| - \dfrac{N}{2} \ln 2 \pi
    \label{eq:hyperparam_max_likelihood}\end{aligned}$$ [@rasmussen_gaussian_2006]. To find the maximum-likelihood estimate of $\bm{\theta}$, namely $\hat{\bm{\theta}} = (\hat{\bm{\mu}}, \hat{\bm{\nu}})$, we may maximize this function subject to the constraint that $\bm{K}$ is positive-semidefinite (positive-semidefiniteness being a necessary property of covariance matrices). The equation for $\ln L_{\Theta}(\bm{\theta}; \bm{y})$ consists of three terms. The first is a measure of fit quality, the second is a complexity penalty, and the third a normalization constant.

The complexity penalty is a function of $\bm{K}$ only, and quantifies the complexity of our attempted fit independent of the data. For a complicated fit, the covariance of any two points is low. Hence the determinant of the covariance matrix $\bm{K}$ is small, and $\ln |\bm{K}|$ diverges with complexity (i.e. as $|\bm{K}| \longrightarrow 0$ so $\ln |\bm{K}| \longrightarrow - \infty$). This strongly penalizes complex models. [For a full discussion see @rasmussen_gaussian_2006 section 5.4.1.] The function $L_{\Theta}$ will in general have multiple maxima, each maximum giving a different tradeoff between 