fit quality and complexity.

The function $r_{\bm{\mu}}$ might be a finite linear combination of basis functions, $(\xi_i)_{i = 1}^N$, and $\bm{\mu}$ the set of coefficients: $$\begin{aligned}
    r(\bm{a}; \bm{\mu}) 
    &= \sum_{i = 1}^{N} \mu_{i} \xi_{i}(\bm{a})\\
    &= \bm{\mu}^{\mathrm{t}} \bm{\xi}(\bm{a})
    \label{eq:gpe_basis}\end{aligned}$$ where $[\bm{\mu}]_{i} = \mu_{i}$ and $[\bm{\xi}(\bm{a})]_{i} = \xi_{i}(\bm{a})$. However, we are free to choose any function for $r$. Specifically we may choose the zero function, such that $r(\bm{a}) = 0$ for all $\bm{a} \in \bm{A}$, whereupon the Gaussian-process fit to the residuals (the second term in equation \[eq:gpe\_mean\]) does all the work of the regression. It is common practice to do this [@rasmussen_gaussian_2006 section 2.7], and we will do so for the rest of this paper. [^3]

The covariance function, $k_{\bm{\nu}}$, is usually chosen from one of a number of standard functions. It is a function of the parameter space $\bm{A}$ and we will need to impose some structure of this space. For our purposes, $\bm{A} = \bm{R}^D$, treated as linear vector space of dimension $D$, together with a pseudo-metric, $d: \bm{A}^2 \longrightarrow \bm{R}$, i.e. a symmetric positive-semidefinite function of two variables that obeys the triangle inequality. [^4] It is common to assume that the pseudo-metric may be written in quadratic form, 