  Methods for analyzing or learning from "fuzzy data" have attracted increasing
attention in recent years. In many cases, however, existing methods (for
precise, non-fuzzy data) are extended to the fuzzy case in an ad-hoc manner,
and without carefully considering the interpretation of a fuzzy set when being
used for modeling data. Distinguishing between an ontic and an epistemic
interpretation of fuzzy set-valued data, and focusing on the latter, we argue
that a "fuzzification" of learning algorithms based on an application of the
generic extension principle is not appropriate. In fact, the extension
principle fails to properly exploit the inductive bias underlying statistical
and machine learning methods, although this bias, at least in principle, offers
a means for "disambiguating" the fuzzy data. Alternatively, we therefore
propose a method which is based on the generalization of loss functions in
empirical risk minimization, and which performs model identification and data
disambiguation simultaneously. Elaborating on the fuzzification of specific
types of losses, we establish connections to well-known loss functions in
regression and classification. We compare our approach with related methods and
illustrate its use in logistic regression for binary classification.
