On a class of interval predictor models with universal reliability

An Interval Predictor Model (IPM) is a rule by which some observable variables (system inputs) are mapped into an interval that is used to predict an inaccessible variable (system output). IPMs have been studied in Campi et al. (2009), where the problem of fitting an IPM on a set of observations has been considered. In the same paper, upper-bounds on the probability that a future system output will fall outside the predicted interval (misprediction) have also been derived in a stationary and independent framework. While these bounds have the notable property of being valid independently of the unknown mechanism that has generated the data, in general the actual probability distribution of the misprediction does depend on the data generation mechanism and, hence, these bounds may introduce conservatism when applied to a specific case. In this paper, we study the reliability of an important class of IPMs, called minimax layers, and show that this class exhibits the special property that the probability distribution of the misprediction is known exactly and is universal, i.e., is always the same irrespective of the data generation mechanism. This result carries important consequences on the use of minimax layers in practice.