= {\operatorname{cov}}(\bm{a},
\bm{a}')$. If a process, $Z$, is Gaussian we write $$\begin{aligned}
  Z \sim \mathrm{GP}(m(\bm{a}), k(\bm{a}, \bm{a}')).\end{aligned}$$

Recall that in curve-fitting (or, *regression* as it is called in the statistical literature) we have a random variable $\bm{A}$ (called the independent variable) and a random variable $Y$ (called the dependent variable), and wish to find the relationship between them. Given *training data* $((\bm{a}_{i}, y(\bm{a}_{i})))_{i = 1}^N$ we may always write the relationship in the form $$\begin{aligned}
    Y(\bm{a}_i) = r(\bm{a}_i) + E(\bm{a}_i)
    \label{eq:regression_model}\end{aligned}$$ where the *regression function* is $r(\bm{a}) := {\operatorname{E}}(Y(\bm{a}))$ for all $\bm{a}$ [@wasserman_all_2007], and $E(\bm{a}_i)$ is a random variable (the *noise*.

We seek an estimator for $r$, denoted $\hat{r}$. In *parametric* regression we assume that this estimator is an element of some parameterized set of functions, $\hat{r} \in \{f_{\bm{\beta}} \,|\, \bm{\beta} \in
\bm{B}\}$. We call this set the *regression model* and the set $\bm{B}$ the set of *regression parameters*. For example, in linear regression we assume that this model is $\{\beta_{0} + \sum_{i = 1}^{D} \beta_{i} a_{i}
\,|\, \beta_{0}, \ldots, \beta_{D} \in \bm{R}\}$. Regression is *non-parametric* if we may not write the estimator in this way. Non-parametric regression is useful when we have no motivation for a parameterized expression for 