# Probabilistic Neural Networks (PNNs) with t-Distributed Outputs: Adaptive Prediction Intervals Beyond Gaussian Assumptions

arXiv: https://arxiv.org/abs/2503.12354 · HTML: https://mlsanity.com/paper/2503.12354/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 47 references that are in the map.
- First posted on arXiv 16 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [On the Pitfalls of Heteroscedastic Uncertainty Estimation with Probabilistic Neural Networks](https://mlsanity.com/paper/2203.09168/) (2022)
- [Torch.manual_seed(3407) is all you need: On the influence of random seeds in deep learning architectures for computer vision](https://mlsanity.com/paper/2109.08203/) (2021)
- [A Survey on Uncertainty Quantification Methods for Deep Learning](https://mlsanity.com/paper/2302.13425/) (2023)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [Faithful Heteroscedastic Regression with Neural Networks](https://mlsanity.com/paper/2212.09184/) (2022)
- [A Comprehensive Survey of Regression Based Loss Functions for Time Series Forecasting](https://mlsanity.com/paper/2211.02989/) (2022)
- [Single-model uncertainty quantification in neural network potentials does not consistently outperform model ensembles](https://mlsanity.com/paper/2305.01754/) (2023)
- [Valid prediction intervals for regression problems](https://mlsanity.com/paper/2107.00363/) (2021)
- [Optimal Training of Mean Variance Estimation Neural Networks](https://mlsanity.com/paper/2302.08875/) (2023)
- [Learning from Students: Applying t-Distributions to Explore Accurate and Efficient Formats for LLMs](https://mlsanity.com/paper/2405.03103/) (2024)
- [Uncertainty Quantification Metrics for Deep Regression](https://mlsanity.com/paper/2405.04278/) (2024)
- [Understanding Square Loss in Training Overparametrized Neural Network Classifiers](https://mlsanity.com/paper/2112.03657/) (2021)
- [One step closer to unbiased aleatoric uncertainty estimation](https://mlsanity.com/paper/2312.10469/) (2023)
- [A Benchmark on Uncertainty Quantification for Deep Learning Prognostics](https://mlsanity.com/paper/2302.04730/) (2023)
- [Outlier-Detection for Reactive Machine Learned Potential Energy Surfaces](https://mlsanity.com/paper/2402.17686/) (2024)
- [Adaptive Activation Functions for Predictive Modeling with Sparse Experimental Data](https://mlsanity.com/paper/2402.05401/) (2024)
- [Two-Stage Surrogate Modeling for Data-Driven Design Optimization with Application to Composite Microstructure Generation](https://mlsanity.com/paper/2401.02008/) (2024)
- [Probabilistic Neural Networks (PNNs) for Modeling Aleatoric Uncertainty in Scientific Machine Learning](https://mlsanity.com/paper/2402.13945/) (2024)

## Built on by (most cited first, 1 of 1)

- [FruitProm: Probabilistic Maturity Estimation and Detection of Fruits and Vegetables](https://mlsanity.com/paper/2510.24885/) (2025)

## Abstract (arXiv)

Traditional neural network regression models provide only point estimates, failing to capture predictive uncertainty. Probabilistic neural networks (PNNs) address this limitation by producing output distributions, enabling the construction of prediction intervals. However, the common assumption of Gaussian output distributions often results in overly wide intervals, particularly in the presence of outliers or deviations from normality. To enhance the adaptability of PNNs, we propose t-Distributed Neural Networks (TDistNNs), which generate t-distributed outputs, parameterized by location, scale, and degrees of freedom. The degrees of freedom parameter allows TDistNNs to model heavy-tailed predictive distributions, improving robustness to non-Gaussian data and enabling more adaptive uncertainty quantification. We incorporate a likelihood based on the t-distribution into neural network training and derive efficient gradient computations for seamless integration into deep learning frameworks. Empirical evaluations on synthetic and real-world data demonstrate that TDistNNs improve the balance between coverage and interval width. Notably, for identical architectures, TDistNNs consistently produce narrower prediction intervals than Gaussian-based PNNs while maintaining proper coverage. This work contributes a flexible framework for uncertainty estimation in neural networks tasked with regression, particularly suited to settings involving complex output distributions.