# A Novel Bayesian Linear Regression Model for the Analysis of Neuroimaging Data
## 3 Methods
### Experimental Setup  
MRI data were standardized to zero mean and unitary standard deviation. As we have a reduced number of subjects (only 53 samples), we have used a Leave-One-Out (LOO) framework to evaluate the model performance. This way, we have trained as many models as available samples, using in each training partition all the subjects except one, which was used afterwards for testing. Then, to evaluate the model performance, we present the results in terms of average accuracy, that is, the percentage of correctly classified test subjects computed over all LOO iterations. Furthermore, since the performance of some methods depends on their initialization we repeated the LOO process 10 times (with different initializations) and depicted the average accuracy over them in order to obtain more statically significant results.  
To complete the performance analysis, the result table includes the final number of voxels selected by each model (and their percentage with respect to the total), computed as the average number of voxels used by each model for each LOO iteration and each run.  
Regarding the different models under study, we considered several configurations to carry out a more comprehensive analysis and more adequate evaluation of the different methods.  
For GPs, we have considered two versions: (1) the standard GP with a linear kernel, denoted as GP, and (2) the previous GP but including ARD capabilities and an FS stage. That is, we first trained a GP with ARD and analyzed the ARD coefficients to select the most relevant features, and then trained a standard GP with the chosen features. Thus, this two-step approach provided a GP with FS capabilities, denoted as GP+FS. For this pruning, we selected the 25% most relevant features in order to compare the performance of this method with DBL-FS. In addition, as both DBL-FS and GPs were formulated for regression problems and our predictive task is a binary classification (0 or 1), we set the threshold to 0.5.  
We have implemented two different approaches for SVMs: (1) a standard SVM with linear kernel and (2) an SVM with a Multi-Kernel Learning (MKL) strategy, denoted as MKL-SVM. In the latter case, we independently considered the different tissues (GM, CSF, and WM) and a different linear kernels for each of them, and subsequently, the model learned the combination of these three kernels, including two parameters for their combination. These parameters were defined as scalars multiplying each kernel term, and a subsequent inner LOO was used to find their optimal values. Thus, the defined combinations coefficients gave more or less relevance to each kernel (therefore, to each tissue), providing additional flexibility to the model.  
For KSSHIBA, we have included two versions, similarly to what was done in GP: (1) the standard KSSHIBA model and (2) a two-stage version of KSSHIBA (denoted as KSSHIBA+FS), in which KSSHIBA was first trained with ARD functionality, and subsequently, we selected the most relevant features to train the model using this subset of features. For these experiments, we initially had 1000 latent factors, from which the model will automatically prune the irrelevant ones. For FS, we kept the highest 25% of voxels equivalently to the number of selected features from the DBL-FS model.  
Finally, we have also defined two approaches for the DBL-FS model, with and without expert knowledge. In the latter one, we have equally initialized the ARD prior for all voxels, setting the parameters \(a\) and \(b\) of random variable \(\alpha\) to 2 and 1, respectively. In the expert knowledge case (denoted as DBL-FS+EK), we have initialized the parameters \(a\) and \(b\) in such a way that the areas of the prefrontal cortex, ventral hippocampus, and lateral ventricles (which are known to be more intensely affected [23]) had more relevance than the rest. In particular, parameter \(a\) was set to 50 and parameter \(b\) was fixed to either 1 or 0.001, depending on whether the voxel belonged to the indicated ROIs or not.