# Confident Neural Network Regression with Bootstrapped Deep Ensembles

arXiv: https://arxiv.org/abs/2202.10903 · HTML: https://mlsanity.com/paper/2202.10903/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 22 Feb 2022.

## 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)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [Concrete Dropout](https://mlsanity.com/paper/1705.07832/) (2017)
- [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://mlsanity.com/paper/2002.06470/) (2020)
- [Evaluating Scalable Bayesian Deep Learning Methods for Robust Computer Vision](https://mlsanity.com/paper/1906.01620/) (2019)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [Uncertainty in Neural Networks: Approximately Bayesian Ensembling](https://mlsanity.com/paper/1810.05546/) (2018)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [Estimating Risk and Uncertainty in Deep Reinforcement Learning](https://mlsanity.com/paper/1905.09638/) (2019)
- [How to Evaluate Uncertainty Estimates in Machine Learning for Regression?](https://mlsanity.com/paper/2106.03395/) (2021)
- [Tight Prediction Intervals Using Expanded Interval Minimization](https://mlsanity.com/paper/1806.11222/) (2018)
- [Prediction intervals for Deep Neural Networks](https://mlsanity.com/paper/2010.04044/) (2020)

## Built on by (most cited first, 3 of 3)

- [Position: Epistemic uncertainty estimation methods are fundamentally incomplete](https://mlsanity.com/paper/2505.23506/) (2025)
- [Deep interpretable ensembles](https://mlsanity.com/paper/2205.12729/) (2022)
- [Likelihood-ratio-based confidence intervals for neural networks](https://mlsanity.com/paper/2308.02221/) (2023)

## Abstract (arXiv)

With the rise of the popularity and usage of neural networks, trustworthy uncertainty estimation is becoming increasingly essential. One of the most prominent uncertainty estimation methods is Deep Ensembles (Lakshminarayanan et al., 2017) . A classical parametric model has uncertainty in the parameters due to the fact that the data on which the model is build is a random sample. A modern neural network has an additional uncertainty component since the optimization of the network is random. Lakshminarayanan et al. (2017) noted that Deep Ensembles do not incorporate the classical uncertainty induced by the effect of finite data. In this paper, we present a computationally cheap extension of Deep Ensembles for the regression setting, called Bootstrapped Deep Ensembles, that explicitly takes this classical effect of finite data into account using a modified version of the parametric bootstrap. We demonstrate through an experimental study that our method significantly improves upon standard Deep Ensembles