Additive regression models with interactions are widely studied in the literature, using methods such as splines or Gaussian process regression. However, these methods can pose challenges for estimation and model selection, due to the presence of many smoothing parameters and the lack of suitable criteria. We propose to address these challenges by extending the I-prior methodology (Bergsma, 2020) to multiple covariates, which may be multidimensional. The I-prior methodology has some advantages over other methods, such as Gaussian process regression and Tikhonov regularization, both theoretically and practically. In particular, the I-prior is a proper prior, is based on minimal assumptions, yields an admissible posterior mean, and estimation of the scale (or smoothing) parameters can be done using an EM algorithm with simple E and M steps. Moreover, we introduce a parsimonious specification of models with interactions, which has two benefits: (i) it reduces the number of scale parameters and thus facilitates the estimation of models with interactions, and (ii) it enables straightforward model selection (among models with different interactions) based on the marginal likelihood.

翻译：包含交互作用的可加回归模型在文献中被广泛研究，其采用的方法包括样条法或高斯过程回归。然而，由于存在大量平滑参数且缺乏合适的准则，这些方法在估计和模型选择方面可能面临挑战。我们提出通过将I先验方法（Bergsma, 2020）扩展至多协变量（这些协变量可能是多维的）来应对这些挑战。在理论和实践层面，I先验方法相较于高斯过程回归和吉洪诺夫正则化等其他方法均具有优势。具体而言，I先验是一种具有良好性质的先验、基于最小假设、可产生可容许的后验均值，并且尺度（或平滑）参数可通过具有简单E步和M步的EM算法进行估计。此外，我们引入了一种简约的交互作用模型规范，其具有两大优势：（i）减少尺度参数数量，从而简化含交互作用模型的估计；（ii）能够基于边际似然实现直观的模型选择（在不同交互作用的模型之间）。