# Ensembles for Uncertainty Estimation: Benefits of Prior Functions and Bootstrapping

arXiv: https://arxiv.org/abs/2206.03633 · HTML: https://mlsanity.com/paper/2206.03633/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 40 references that are in the map.
- First posted on arXiv 8 Jun 2022.

## Builds on (most cited first, 23 of 23)

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [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)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [Deep Exploration via Bootstrapped DQN](https://mlsanity.com/paper/1602.04621/) (2016)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [A Tutorial on Thompson Sampling](https://mlsanity.com/paper/1707.02038/) (2017)
- [Randomized Prior Functions for Deep Reinforcement Learning](https://mlsanity.com/paper/1806.03335/) (2018)
- [Epistemic Neural Networks](https://mlsanity.com/paper/2107.08924/) (2021)
- [Repulsive Deep Ensembles are Bayesian](https://mlsanity.com/paper/2106.11642/) (2021)
- [Bayesian Deep Ensembles via the Neural Tangent Kernel](https://mlsanity.com/paper/2007.05864/) (2020)
- [Ensemble Sampling](https://mlsanity.com/paper/1705.07347/) (2017)
- [DICE: Diversity in Deep Ensembles via Conditional Redundancy Adversarial Estimation](https://mlsanity.com/paper/2101.05544/) (2021)
- [Hypermodels for Exploration](https://mlsanity.com/paper/2006.07464/) (2020)
- [Maximizing Overall Diversity for Improved Uncertainty Estimates in Deep Ensembles](https://mlsanity.com/paper/1906.07380/) (2019)
- [From Predictions to Decisions: The Importance of Joint Predictive Distributions](https://mlsanity.com/paper/2107.09224/) (2021)
- [Diversity Matters When Learning From Ensembles](https://mlsanity.com/paper/2110.14149/) (2021)
- [An Analysis of Ensemble Sampling](https://mlsanity.com/paper/2203.01303/) (2022)
- [Beyond Marginal Uncertainty: How Accurately can Bayesian Regression Models Estimate Posterior Predictive Correlations?](https://mlsanity.com/paper/2011.03178/) (2020)
- [Diversity regularization in deep ensembles](https://mlsanity.com/paper/1802.07881/) (2018)
- [Evaluating High-Order Predictive Distributions in Deep Learning](https://mlsanity.com/paper/2202.13509/) (2022)

## Built on by (most cited first, 19 of 19)

- [To Believe or Not to Believe Your LLM](https://mlsanity.com/paper/2406.02543/) (2024)
- [Uncertainty quantification in fine-tuned LLMs using LoRA ensembles](https://mlsanity.com/paper/2402.12264/) (2024)
- [Leveraging Demonstrations to Improve Online Learning: Quality Matters](https://mlsanity.com/paper/2302.03319/) (2023)
- [Using AI Uncertainty Quantification to Improve Human Decision-Making](https://mlsanity.com/paper/2309.10852/) (2023)
- [Probabilistic Inference in Reinforcement Learning Done Right](https://mlsanity.com/paper/2311.13294/) (2023)
- [Efficient Exploration via Epistemic-Risk-Seeking Policy Optimization](https://mlsanity.com/paper/2302.09339/) (2023)
- [Neural Exploitation and Exploration of Contextual Bandits](https://mlsanity.com/paper/2305.03784/) (2023)
- [Online Bandit Learning with Offline Preference Data for Improved RLHF](https://mlsanity.com/paper/2406.09574/) (2024)
- [Bayesian posterior approximation with stochastic ensembles](https://mlsanity.com/paper/2212.08123/) (2022)
- [Best Policy Learning from Trajectory Preference Feedback](https://mlsanity.com/paper/2501.18873/) (2025)
- [Satisficing Exploration for Deep Reinforcement Learning](https://mlsanity.com/paper/2407.12185/) (2024)
- [Fully Heteroscedastic Count Regression with Deep Double Poisson Networks](https://mlsanity.com/paper/2406.09262/) (2024)
- [Response Uncertainty and Probe Modeling: Two Sides of the Same Coin in LLM Interpretability?](https://mlsanity.com/paper/2505.18575/) (2025)
- [Bayesian Quadrature for Neural Ensemble Search](https://mlsanity.com/paper/2303.08874/) (2023)
- [Likelihood-ratio-based confidence intervals for neural networks](https://mlsanity.com/paper/2308.02221/) (2023)
- [Epinet for Content Cold Start](https://mlsanity.com/paper/2412.04484/) (2024)
- [Pointwise confidence estimation in the non-linear ℓ2-regularized least squares](https://mlsanity.com/paper/2506.07088/) (2025)
- [Sharp analysis of linear ensemble sampling](https://mlsanity.com/paper/2602.08026/) (2026)
- [Trends, Challenges, and Future Directions in Deep Learning for Glaucoma: A Systematic Review](https://mlsanity.com/paper/2411.05876/) (2024)

## Abstract (arXiv)

In machine learning, an agent needs to estimate uncertainty to efficiently explore and adapt and to make effective decisions. A common approach to uncertainty estimation maintains an ensemble of models. In recent years, several approaches have been proposed for training ensembles, and conflicting views prevail with regards to the importance of various ingredients of these approaches. In this paper, we aim to address the benefits of two ingredients -- prior functions and bootstrapping -- which have come into question. We show that prior functions can significantly improve an ensemble agent's joint predictions across inputs and that bootstrapping affords additional benefits if the signal-to-noise ratio varies across inputs. Our claims are justified by both theoretical and experimental results.