This paper suggests a modification of the Conformal Prediction framework for regression that will strengthen the associated guarantee of validity. We motivate the need for this modification and argue that our conformal regressors are more closely tied to the actual error distribution of the underlying model, thus allowing for more natural interpretations of the prediction intervals. In the experimentation, we provide an empirical comparison of our conformal regressors to traditional conformal regressors and show that the proposed modification results in more robust two-tailed predictions, and more efficient one-tailed predictions.

Conformal classifiers output confidence prediction regions, i.e., multi-valued predictions that are guaranteed to contain the true output value of each test pattern with some predefined probability. In order to fully utilize the predictions provided by a conformal classifier, it is essential that those predictions are reliable, i.e., that a user is able to assess the quality of the predictions made. Although conformal classifiers are statistically valid by default, the error probability of the prediction regions output are dependent on their size in such a way that smaller, and thus potentially more interesting, predictions are more likely to be incorrect. This paper proposes, and evaluates, a method for producing refined error probability estimates of prediction regions, that takes their size into account. The end result is a binary conformal confidence predictor that is able to provide accurate error probability estimates for those prediction regions containing only a single class label.

Conformal prediction is a learning framework that produces models that associate with each of their predictions a measure of statistically valid confidence. These models are typically constructed on top of traditional machine learning algorithms. An important result of conformal prediction theory is that the models produced are provably valid under relatively weak assumptions—in particular, their validity is independent of the specific underlying learning algorithm on which they are based. Since validity is automatic, much research on conformal predictors has been focused on improving their informational and computational efficiency. As part of the efforts in constructing efficient conformal predictors, aggregated conformal predictors were developed, drawing inspiration from the field of classification and regression ensembles. Unlike early definitions of conformal prediction procedures, the validity of aggregated conformal predictors is not fully understood—while it has been shown that they might attain empirical exact validity under certain circumstances, their theoretical validity is conditional on additional assumptions that require further clarification. In this paper, we show why validity is not automatic for aggregated conformal predictors, and provide a revised definition of aggregated conformal predictors that gains approximate validity conditional on properties of the underlying learning algorithm.

In this paper, we propose a practically useful means of interpreting the predictions produced by a conformal classifier. The proposed interpretation leads to a classifier with a reject option, that allows the user to limit the number of erroneous predictions made on the test set, without any need to reveal the true labels of the test objects. The method described in this paper works by estimating the cumulative error count on a set of predictions provided by a conformal classifier, ordered by their confidence. Given a test set and a user-specified parameter k, the proposed classification procedure outputs the largest possible amount of predictions containing on average at most k errors, while refusing to make predictions for test objects where it is too uncertain. We conduct an empirical evaluation using benchmark datasets, and show that we are able to provide accurate estimates for the error rate on the test set.

In this paper, we study a generalization of a recently developed strategy for generating conformal predictor ensembles: out-of-bag calibration. The ensemble strategy is evaluated, both theoretically and empirically, against a commonly used alternative ensemble strategy, bootstrap conformal prediction, as well as common non-ensemble strategies. A thorough analysis is provided of out-of-bag calibration, with respect to theoretical validity, empirical validity (error rate), efficiency (prediction region size) and p-value stability (the degree of variance observed over multiple predictions for the same object). Empirical results show that out-of-bag calibration displays favorable characteristics with regard to these criteria, and we propose that out-of-bag calibration be adopted as a standard method for constructing conformal predictor ensembles.