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.

翻译：完全贝叶斯显著性检验（FBST）具有诸多理想特性，例如无需假设具有非零先验概率，同时能为$H_0$提供证据度量。然而，将FBST应用于非参数场景的尝试仍然有限，其主要障碍在于需要在函数空间中获取最高后验密度（HPD）。本研究采用高斯过程，为如下形式的假设提供解析可处理的FBST：$$ H_0: g(\boldsymbol{x}) = \boldsymbol{b}(\boldsymbol{x})\boldsymbol{\beta}, \quad \forall \boldsymbol{x} \in \mathcal{X}, \quad \boldsymbol{\beta} \in \mathbb{R}^k, $$ 其中$g(\cdot)$为回归函数，$\boldsymbol{b}(\cdot)$是线性无关的线性函数向量（例如$\boldsymbol{b}(\boldsymbol{x}) = \boldsymbol{x}'$），$\mathcal{X}$为协变量域。我们还利用实用假设来验证线性模型的拟合是否可能近似成立而非严格精确，从而纳入测量误差与效用判断等有价值信息。此项研究拓展了FBST的理论框架，使其能够应用于非参数场景，并提供了一种可轻松检验线性模型对数据适用性、且能自动执行变量选择的分析流程。