e rest of the data $\mathbf{y}_{-j}$. These likelihoods are used to monitor the predictive performance of the model. This performance is used to estimate the generalization error, and it can be used to carry out model selection [@kohavi1995study; @Rasmussen2006; @vehtari2017practical].

The Bayesian CV estimate of the predictive fit with given parameters ${\boldsymbol{\varphi}}$ is $$\mbox{CV} = \sum_{j=1}^n
    \log p(\mathbf{y}_j \mid \mathbf{y}_{-j},{\boldsymbol{\varphi}}),$$ where $p(\mathbf{y}_j \mid \mathbf{y}_{-j},{\boldsymbol{\varphi}})$ is the predictive likelihood of the data $\mathbf{y}_j$ given the rest of the data. The best parameter values with respect to CV can be computed by enumerating the possible parameter values and selecting the one which gives the best fit in terms of CV.

Experimental results {#sec:expResults}
====================

In this section, we present numerical results using the GP model for limited x-ray tomography problems. All the computations were implemented in <span style="font-variant:small-caps;">Matlab</span> 9.4 (R2018a) and performed on an Intel Core i5 at 2.3 GHz and CPU 8GB 2133MHz LPDDR3 memory.

For both simulated data (see Section \[SimulatedData\]) and real data (see Section \[RealData\]) we use $m=10^4$ basis functions in . The measurements are obtained from the line integral of each x-ray over the attenuation coefficient of the measured objects. The measurements are taken for each direction (angle of view), and later they will be referred to as projections. The same number of rays in each direction is used. The computation of the hyperparameters is carried out using the Metropolis–Hastings algorithms with $5\,000$ samples, and the first $1\,000$ samples are thrown away ([*burn-in*]{} period). The reconstruction is computed by taking the conditional mean of the object estimate.

Simulated data: 2D Chest phantom {#SimulatedData}
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As for the simulated data, we use one slice of <span style="font-variant:small-caps;">Matlab</span>�