Nonparametric FBST for Validating Linear Models

The Full Bayesian Significance Test (FBST) possesses many desirable aspects, such as dismissing the need for hypotheses to have positive prior probability and providing a measure of evidence against $H_0$. Still, few attempts have been made to bring the FBST to nonparametric settings, with the main drawback being the need to obtain the highest posterior density (HPD) in a function space. In this work, we use a Gaussian processes prior to derive the FBST for hypotheses of the type $$H_0:g(\mathit{x})=\mathit{b}(\mathit{x})\mathit{\beta },\mathrm{\forall }\mathit{x}\in \mathcal{X},\mathit{\beta }\in {\mathbb{R}}^k,$$ where $g(\cdot )$ is the regression function, $\mathit{b}(\cdot )$ is a vector of linearly independent linear functions – such as $\mathit{b}(\mathit{x})={\mathit{x}}^{\mathrm{\prime }}$ – and $\mathcal{X}$ is the covariates’ domain. We also make use of pragmatic hypotheses to verify if the data might be compatible with a linear model when factors such as measurement errors or utility judgments are accounted for. This contribution extends the theory of the FBST, allowing its application in nonparametric settings and providing a procedure that easily tests if linear models are adequate for the data and that can automatically perform variable selection.