parameter space, where the model is constrained by data on one side only. We may consider a point to be near the boundary of parameter space if it is within one parameter length scale of it.

We may in fact use LOOCV to optimize the hyperparameter vector, as opposed to the maximum-likelihood method described in section \[sec:gpe\]. Note that the logarithm of the likelihood of the hyperparameter vector when leaving out the $i$-th pair of data (called the *leave-one-out likelihood*) is $$\begin{aligned}
    \ln L_{-i, \bm{\Theta}}(\bm{\theta}; \bm{y}) = - \dfrac{1}{2} \ln \sigma_{-i}^2(\bm{a}_i) - \dfrac{1}{2} e_{-i}^2(\bm{a}_i) - \dfrac{1}{2} \ln 2 \pi\end{aligned}$$ [@rasmussen_gaussian_2006]. Hence the logarithm of the joint leave-one-out likelihood (LOO likelihood) is $$\begin{aligned}
    \ln L_{\mathrm{LOO}, \bm{\Theta}} = \sum_{i = 1}^N \ln L_{\mathrm{pseudo}, \bm{\Theta}}(\bm{\theta}; \bm{y}),\end{aligned}$$ which we may maximize to give the *maximum LOO likelihood estimate* of the hyperparameter vector. This estimate is more robust to mispecification of the covariance function [@bachoc_cross_2013] than the maximum-likelihood estimate (equation \[eq:hyperparam\_max\_likelihood\]).

Conditioning the likelihood {#sec:conditioning}
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If the estimator fails validation then the covariance function has been misspecified. In this case we may do one of two things: choose a different covariance function, or transform the data so that the function is better suited to 