Distribution-Free Inference for the Regression Function of Binary Classification

One of the key objects of binary classification is the regression function, i.e., the conditional expectation of the class labels given the inputs. With the regression function not only a Bayes optimal classifier can be defined, but it also encodes the corresponding misclassification probabilities. The paper presents a resampling framework to construct exact, distribution-free and non-asymptotically guaranteed confidence regions for the true regression function for any user-chosen confidence level. Then, specific algorithms are suggested to demonstrate the framework. It is proved that the constructed confidence regions are strongly consistent, that is, any false model is excluded in the long run with probability one. The exclusion is quantified with probably approximately correct type bounds, as well. Finally, the algorithms are validated via numerical experiments, and the methods are compared to approximate asymptotic confidence ellipsoids.