# Easy Uncertainty Quantification (EasyUQ): Generating Predictive Distributions from Single-valued Model Output

arXiv: https://arxiv.org/abs/2212.08376 · HTML: https://mlsanity.com/paper/2212.08376/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2022.

## Built on by (most cited first, 11 of 11)

- [Uncertainty quantification for data-driven weather models](https://mlsanity.com/paper/2403.13458/) (2024)
- [Probabilistic neural operators for functional uncertainty quantification](https://mlsanity.com/paper/2502.12902/) (2025)
- [EnScale: Temporally-consistent multivariate generative downscaling via proper scoring rules](https://mlsanity.com/paper/2509.26258/) (2025)
- [Postprocessing of point predictions for probabilistic forecasting of day-ahead electricity prices: The benefits of using isotonic distributional regression](https://mlsanity.com/paper/2404.02270/) (2024)
- [Probabilistic measures afford fair comparisons of AIWP and NWP model output](https://mlsanity.com/paper/2506.03744/) (2025)
- [On the Predictive Skill of Artificial Intelligence-based Weather Models for Extreme Events using Uncertainty Quantification](https://mlsanity.com/paper/2511.17176/) (2025)
- [Machine learning models for daily rainfall forecasting in Northern Tropical Africa using tropical wave predictors](https://mlsanity.com/paper/2408.16349/) (2024)
- [Data-driven rainfall prediction at a regional scale: a case study with Ghana](https://mlsanity.com/paper/2410.14062/) (2024)
- [Statistical post-processing yields accurate probabilistic forecasts from Artificial Intelligence weather models](https://mlsanity.com/paper/2504.12672/) (2025)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)
- [Suitable Measures for the Potential Operational Utility of AI NWP Rainfall Forecasts Over Africa](https://mlsanity.com/paper/2609.31775/) (2026)

## Abstract (arXiv)

How can we quantify uncertainty if our favorite computational tool - be it a numerical, a statistical, or a machine learning approach, or just any computer model - provides single-valued output only? In this article, we introduce the Easy Uncertainty Quantification (EasyUQ) technique, which transforms real-valued model output into calibrated statistical distributions, based solely on training data of model output-outcome pairs, without any need to access model input. In its basic form, EasyUQ is a special case of the recently introduced Isotonic Distributional Regression (IDR) technique that leverages the pool-adjacent-violators algorithm for nonparametric isotonic regression. EasyUQ yields discrete predictive distributions that are calibrated and optimal in finite samples, subject to stochastic monotonicity. The workflow is fully automated, without any need for tuning. The Smooth EasyUQ approach supplements IDR with kernel smoothing, to yield continuous predictive distributions that preserve key properties of the basic form, including both, stochastic monotonicity with respect to the original model output, and asymptotic consistency. For the selection of kernel parameters, we introduce multiple one-fit grid search, a computationally much less demanding approximation to leave-one-out cross-validation. We use simulation examples and forecast data from weather prediction to illustrate the techniques. In a study of benchmark problems from machine learning, we show how EasyUQ and Smooth EasyUQ can be integrated into the workflow of neural network learning and hyperparameter tuning, and find EasyUQ to be competitive with conformal prediction, as well as more elaborate input-based approaches.