Nonparametric FBST for Validating Linear Models

In Bayesian analysis, testing for linearity requires placing a prior to the entire space of potential regression functions. This poses a problem for many standard tests, as assigning positive prior probability to such a hypothesis is challenging. The Full Bayesian Significance Test (FBST) sidesteps this issue, standing out for also being logically coherent and offering a measure of evidence against <inline-formula> <mml:math id="mm1"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula>, although its application to nonparametric settings is still limited. In this work, we use Gaussian process priors to derive FBST procedures that evaluate general linearity assumptions, such as testing the adherence of data and performing variable selection to linear models. 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 for its application in nonparametric settings and requiring, at most, simple optimization procedures to reach the desired conclusion.