# Uncertainty propagation through trained multi-layer perceptrons: Exact analytical results

arXiv: https://arxiv.org/abs/2601.16830 · HTML: https://mlsanity.com/paper/2601.16830/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 19 references that are in the map.
- First posted on arXiv 23 Jan 2026.

## Builds on (most cited first, 4 of 4)

- [An Analytic Solution to Covariance Propagation in Neural Networks](https://mlsanity.com/paper/2403.16163/) (2024)
- [Uncertainty Propagation in Deep Neural Networks Using Extended Kalman Filtering](https://mlsanity.com/paper/1809.06009/) (2018)
- [Analytical results for uncertainty propagation through trained machine learning regression models](https://mlsanity.com/paper/2404.11224/) (2024)
- [Uncertainty propagation in feed-forward neural network models](https://mlsanity.com/paper/2503.21059/) (2025)

## Abstract (arXiv)

We give analytical results for propagation of uncertainty through trained multi-layer perceptrons (MLPs) with a single hidden layer and ReLU activation functions. More precisely, we give expressions for the mean and variance of the output when the input is multivariate Gaussian. In contrast to previous results, we obtain exact expressions without resort to a series expansion.