Kernel methods has been used in Bayesian inference for construction adaptive quadrature schemes, and/or adaptive proposal densities and/or as possible emulators of noisy and costly posteriors. Furthermore, in Bayesian inference, the choice of prior distributions plays a central role, and a large body of literature has investigated constructions in which priors are linked, either directly or indirectly, to the likelihood function or to the observed data. These approaches are often motivated by the desire to reduce subjectivity in prior specification while retaining coherence with the underlying statistical model. These ``non-informative'' specifications are determined by the model structure rather than by expert knowledge. Some well examples are given by (a) the empirical Bayes approach for prior parameter tuning, (b) Jeffreys priors (which are derived from the Fisher information) and (c) reference priors, to name a few. More generally, reference priors were developed as a formal framework to maximize the expected information gain from data, providing a principled way to construct priors. In this work, we aim to extend the methodologies developed for the so-called partial, intrinsic, and fractional Bayes factors, along with related approaches. We also show the relationship with the use some improper priors and the application of the proposed approach for model selection purposes.