  This paper investigates when the importance weighting (IW) correction is
needed to address covariate shift, a common situation in supervised learning
where the input distributions of training and test data differ. Classic results
show that the IW correction is needed when the model is parametric and
misspecified. In contrast, recent results indicate that the IW correction may
not be necessary when the model is nonparametric and well-specified. We examine
the missing case in the literature where the model is nonparametric and
misspecified, and show that the IW correction is needed for obtaining the best
approximation of the true unknown function for the test distribution. We do
this by analyzing IW-corrected kernel ridge regression, covering a variety of
settings, including parametric and nonparametric models, well-specified and
misspecified settings, and arbitrary weighting functions.
