may be costly to evaluate. In such circumstances we may reduce the computational burden by constructing a cheaper approximation of the likelihood using GPE.

Gaussian-process emulation {#sec:gpe}
==========================

At the heart of GPE [@ohagan_curve_1978; @sacks_design_1989] are random processes, which are sets of random variables. A realization of a random process (i.e. a realization of each individual random variable) is a function, meaning that we may use Gaussian-processes in the context of curve-fitting. More rigorously, a random process, $\{Y(\bm{a})\}_{\bm{a} \in
  \bm{A}}$, is an indexed set of random variables. Again, we call an index, $\bm{a}$, a *parameter* (or *parameter vector* if $\bm{A}$ is multidimensional), and we call the index set, $\bm{A}$, the *parameter space*. A random process is *Gaussian* if every finite subset of the process has a multivariate normal distribution. A process is completely specified by the joint PDF for every finite subset of the process. Since the joint PDF of a multivariate normal distribution is completely specified by its mean and covariance, a Gaussian process is completely specified by its mean and covariance. The mean is the function $m: A \longrightarrow \bm{R}$ such that $m(\bm{a}) = {\operatorname{E}}(Y(\bm{a}))$. The covariance is the function $k:
\bm{A}^2 \longrightarrow \bm{R}$ such that $k(\bm{a}, \bm{a}') 