Uncertainty quantification in prediction presents a compelling challenge with vast applications across various domains, including biomedical science, economics, and weather forecasting. There exists a wide array of methods for constructing prediction intervals, such as quantile regression and conformal prediction. However, practitioners often face the challenge of selecting the most suitable method for a specific real-world data problem. In response to this dilemma, we introduce a novel and universally applicable strategy called Universally Trainable Optimal Predictive Intervals Aggregation (UTOPIA). This technique excels in efficiently aggregating multiple prediction intervals while maintaining a small average width of the prediction band and ensuring coverage. UTOPIA is grounded in linear or convex programming, making it straightforward to train and implement. In the specific case where the prediction methods are elementary basis functions, as in kernel and spline bases, our method becomes the construction of a prediction band. Our proposed methodologies are supported by theoretical guarantees on the coverage probability and the average width of the aggregated prediction interval, which are detailed in this paper. The practicality and effectiveness of UTOPIA are further validated through its application to synthetic data and two real-world datasets in finance and macroeconomics.
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