Class Prior Estimation under Covariate Shift -- no Problem?

We show that in the context of classification the property of source and target distributions to be related by covariate shift may break down when the information content captured in the covariates is reduced, for instance by discretization of the covariates, dropping some of them, or by any transformation of the covariates even if it is domain-invariant. The consequences of this observation for class prior estimation under covariate shift are discussed. A probing algorithm as alternative approach to class prior estimation under covariate shift is proposed.