  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.
