# Prediction Intervals: Split Normal Mixture from Quality-Driven Deep Ensembles

arXiv: https://arxiv.org/abs/2007.09670 · HTML: https://mlsanity.com/paper/2007.09670/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2020.

## Built on by (most cited first, 14 of 14)

- [A Survey on Uncertainty Quantification Methods for Deep Learning](https://mlsanity.com/paper/2302.13425/) (2023)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [PI3NN: Out-of-distribution-aware prediction intervals from three neural networks](https://mlsanity.com/paper/2108.02327/) (2021)
- [Dual Accuracy-Quality-Driven Neural Network for Prediction Interval Generation](https://mlsanity.com/paper/2212.06370/) (2022)
- [Neural Spline Search for Quantile Probabilistic Modeling](https://mlsanity.com/paper/2301.04857/) (2023)
- [Surrogate uncertainty estimation for your time series forecasting black-box: learn when to trust](https://mlsanity.com/paper/2302.02834/) (2023)
- [Tube Loss: A Novel Approach for Prediction Interval Estimation](https://mlsanity.com/paper/2412.06853/) (2024)
- [Can a single neuron learn predictive uncertainty?](https://mlsanity.com/paper/2106.03702/) (2021)
- [Centroid Approximation for Bootstrap: Improving Particle Quality at Inference](https://mlsanity.com/paper/2110.08720/) (2021)
- [Meta-learning to Calibrate Gaussian Processes with Deep Kernels for Regression Uncertainty Estimation](https://mlsanity.com/paper/2312.07952/) (2023)
- [Interval Forecasts for Gas Prices in the Face of Structural Breaks -- Statistical Models vs. Neural Networks](https://mlsanity.com/paper/2407.16723/) (2024)
- [Large width penalization for neural network-based prediction interval estimation](https://mlsanity.com/paper/2411.19181/) (2024)
- [Post-Hoc Split-Point Self-Consistency Verification for Efficient, Unified Quantification of Aleatoric and Epistemic Uncertainty in Deep Learning](https://mlsanity.com/paper/2509.13262/) (2025)
- [Bi-Level Chaotic Fusion Based Graph Convolutional Network for Stock Market Prediction Interval](https://mlsanity.com/paper/2605.16324/) (2026)

## Abstract (arXiv)

Prediction intervals are a machine- and human-interpretable way to represent predictive uncertainty in a regression analysis. In this paper, we present a method for generating prediction intervals along with point estimates from an ensemble of neural networks. We propose a multi-objective loss function fusing quality measures related to prediction intervals and point estimates, and a penalty function, which enforces semantic integrity of the results and stabilizes the training process of the neural networks. The ensembled prediction intervals are aggregated as a split normal mixture accounting for possible multimodality and asymmetricity of the posterior predictive distribution, and resulting in prediction intervals that capture aleatoric and epistemic uncertainty. Our results show that both our quality-driven loss function and our aggregation method contribute to well-calibrated prediction intervals and point estimates.