Nonparametric FBST for Validating Linear Models

Abstract

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 $$H_0:g(\mathbf{x})=\mathbf{b}(\mathbf{x}){\hat{\mathbf{I}}}^2,\forall \mathbf{x}\in \mathcal{X},{\hat{\mathbf{I}}}^2\in {\mathbb{R}}^k,$$ where $g(\cdot )$ is the regression function, $\mathbf{b}(\cdot )$ is a vector of linearly independent linear functions -- such as $\mathbf{b}(\mathbf{x})={\mathbf{x}}^{\prime }$ -- 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.

All code available in https://github.com/rflassance/lmFBST