# A Novel Bayesian Linear Regression Model for the Analysis of Neuroimaging Data
## 5 Discussion  
This study shows, for the first time, the great advantage of using DBL-FS for the detection and characterization of the morphometric brain changes in a rodent model of schizophrenia. This Bayesian model was adapted for neuroimaging data, characterized by a low sample-to-variable ratio (53 samples vs. 1,077,656 voxels in our case) relying on a dual formulation of the Bayesian Linear Regression model. Furthermore, as the main novelty of this proposal, we combine this dual formulation with a prior over the primal weights to learn the feature relevance over the input features, forcing an automatic FS. Finally, we can exploit the Bayesian nature of the model to include specific prior knowledge to guide the learning process and counterweight the limitations caused by the low sample size of the problem. Thus, the comparison in terms of performance with the baselines provides clear evidence of the promising results of the proposed DBL-FS model in the characterization of neuroimaging data in mental disorders. Note that DBL-FS is able to largely outperform the baselines in prediction accuracy, showing an advantage of 5.7% in terms of accuracy over the best baseline. In addition, DBL-FS is the only method capable of detecting regions within the three brain tissues that are known to be relevant in the biology of schizophrenia. In this sense, the relevance learned by the GPs is inconsistent between the different LOO iterations, generating a scattered voxel selection and, therefore, a non-localized, unreliable, and uninterpretable mask. On the other hand, KSSHIBA provides a consistent result but only finds relevant regions within WM tissue and, therefore, ignores relevant regions and includes some irrelevant areas. Analyzing in detail the regions selected by the DBL-FS and DBL-FS+Ek models, we can verify that these areas belong to brain regions whose morphometric changes have been related to schizophrenia, based on the literature. First, as for CSF, the areas with the greatest weight were the most frontal areas of the lateral ventricles and the third ventricle. One of the morphometric hallmarks in schizophrenia is the enlargement of the ventricles [23; 40], which is consistent with the learned selection. Second, regarding GM, our model clearly defined anatomical areas, such as the prefrontal cortex (PFC), hippocampus, amygdala,  
Figure 3: Brain masks obtained by the FS of each model. Colors are defined in a linear scale and associated to the relevance of the voxel (white: more relevant; dark red: less relevant). GP-FS model yields meaningless results in neurobiological terms, where no anatomical regions can be identified. KSSHIBA-FS model only identifies brain areas related to WM deficits in schizophrenia. Both DBL-FS and DBL-FS+EK learn similar relevance in WM, GM, and CSF brain areas of interest in schizophrenia, such as the hippocampus (hipp), prefrontal cortex (PFC), amygdala (Aa), septum, lateral ventricles (LV), corpus callosum (cc), WM cerebellar (WM Cb), and WM brainstem (WM BS) fibers. and septum, some of them in both hemispheres. Numerous studies have demonstrated the relevance of the morphological changes of these areas in mental disorders [41, 42] together with the disconnection and lack of symmetry between both cerebral hemispheres [43, 44]. Similar volumetric abnormalities have also been reported for the animal model used in this study [24, 45]. In addition, the method also detected the medial septum, which plays a significant role in dopamine-related disorders such as schizophrenia [46, 47] and addictions [48, 49, 50], which highlights the relevance of this structure in mental disorders. Regarding WM, the method found three well-defined brain areas, the frontal part of corpus callosum and WM tracts of the brainstem and the cerebellum [51, 52]. Regarding the inclusion of expert knowledge by means of the \(\mathbf{\alpha}\) prior, it reveals two interesting behaviors.