Nonparametric FBST for Validating Linear Models

Abstract: The Full Bayesian Significance Test (FBST) possesses many desirable aspects, such as not requiring a non-zero prior probability for hypotheses while also producing a measure of evidence for $H_0$. 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 Gaussian processes to provide an analytically tractable FBST for hypotheses of the type $$ H_0: g(\boldsymbol{x}) = \boldsymbol{b}(\boldsymbol{x})\boldsymbol{\beta}, \quad \forall \boldsymbol{x} \in \mathcal{X}, \quad \boldsymbol{\beta} \in \mathbb{R}^k, $$ where $g(\cdot)$ is the regression function, $\boldsymbol{b}(\cdot)$ is a vector of linearly independent linear functions -- such as $\boldsymbol{b}(\boldsymbol{x}) = \boldsymbol{x}'$ -- and $\mathcal{X}$ is the covariates' domain. We also make use of pragmatic hypotheses to verify if the adherence of linear models may be approximately instead of exactly true, allowing for the inclusion of valuable information such as measurement errors and utility judgments. 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.