Let $(X,Y)$ be a random variable consisting of an observed feature vector $Xin mathcal{X}$ and an unobserved class label $Yin {1,2,...,L}$ with unknown joint distribution. In addition, let $mathcal{D}$ be a training data set consisting of $n$ completely observed independent copies of $(X,Y)$. Usual classification procedures provide point predictors (classifiers) $widehat{Y}(X,mathcal{D})$ 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 $theta =1,2,...,L$ a p-value $pi_{theta}(X,mathcal{D})$ for the null hypothesis that $Y=theta$, treating $Y$ temporarily as a fixed parameter. In other words, the point predictor $widehat{Y}(X,mathcal{D})$ is replaced with a prediction region for $Y$ with a certain confidence. We argue that (i) this approach is advantageous 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.
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