P-values for classification

Let (X,Y ) be a random variable consisting of an observed feature vector X ∈ X and an unobserved class label Y ∈ {1, 2, . . . , L} with unknown joint distribution. In addition, let D be a training data set consisting of n completely observed independent copies of (X,Y ). Usual classification procedures provide point predictors (classifiers) Ŷ (X,D) of Y or estimate the conditional distribution of Y given X. In order to quantify the certainty of classifying X we propose to construct for each θ = 1, 2, . . . , L a p-value πθ(X,D) for the null hypothesis that Y = θ, treating Y temporarily as a fixed parameter. In other words, the point predictor Ŷ (X,D) is replaced with a prediction region for Y with a certain confidence. We argue that (i) this approach is advantegeous over traditional approaches and (ii) any reasonable classifier can be modified to yield nonparametric p-values. We discuss issues such as optimality, single use and multiple use validity, as well as computational and graphical aspects.