A crucial step in fitting a regression model to data is determining the model's structure, i.e., the subset of explanatory variables to be included. However, the uncertainty in this step is often overlooked due to a lack of satisfactory methods. Frequentists have no broadly applicable confidence set constructions for a model's structure, and Bayesian posterior credible sets do not achieve the desired finite-sample coverage. In this paper, we propose an extension of the possibility-theoretic inferential model (IM) framework that offers reliable, data-driven uncertainty quantification about the unknown model structure. This particular extension allows for the inclusion of incomplete prior information about the unknown structure that facilitates regularization. We prove that this new, regularized, possibilistic IM's uncertainty quantification is suitably calibrated relative to the set of joint distributions compatible with the data-generating process and assumed partial prior knowledge about the structure. This implies, among other things, that the derived confidence sets for the unknown model structure attain the nominal coverage probability in finite samples. We provide background and guidance on quantifying prior knowledge in this new context and analyze two benchmark data sets, comparing our results to those obtained by existing methods.

翻译：在将回归模型拟合至数据时，确定模型结构（即待纳入的解释变量子集）是一个关键步骤。然而，由于缺乏令人满意的方法，这一步骤中的不确定性常被忽视。频率学派尚未建立适用于广泛场景的模型结构置信集构造方法，而贝叶斯后验可信集亦无法达到所需的有限样本覆盖水平。本文提出了一种可能性理论推断模型（IM）框架的扩展，旨在为未知模型结构提供可靠的数据驱动不确定性量化。该扩展特别允许纳入关于未知结构的不完整先验信息，从而促进正则化。我们证明，这种新型正则化可能性IM的不确定性量化相对于与数据生成过程及所假设的结构部分先验知识相容的联合分布集合是适当校准的。这意味着，所导出的未知模型结构置信集在有限样本中达到名义覆盖概率。我们在此新背景下提供了量化先验知识的背景与指导，并分析了两组基准数据集，将所得结果与现有方法进行比较。