  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.
