# Density-aware NeRF Ensembles: Quantifying Predictive Uncertainty in Neural Radiance Fields

arXiv: https://arxiv.org/abs/2209.08718 · HTML: https://mlsanity.com/paper/2209.08718/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 29 references that are in the map.
- First posted on arXiv 19 Sep 2022.

## Builds on (most cited first, 25 of 25)

- [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)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [TensoRF: Tensorial Radiance Fields](https://mlsanity.com/paper/2203.09517/) (2022)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [NICE-SLAM: Neural Implicit Scalable Encoding for SLAM](https://mlsanity.com/paper/2112.12130/) (2021)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [iMAP: Implicit Mapping and Positioning in Real-Time](https://mlsanity.com/paper/2103.12352/) (2021)
- [Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age](https://mlsanity.com/paper/1606.05830/) (2016)
- [Vision-Only Robot Navigation in a Neural Radiance World](https://mlsanity.com/paper/2110.00168/) (2021)
- [Dex-NeRF: Using a Neural Radiance Field to Grasp Transparent Objects](https://mlsanity.com/paper/2110.14217/) (2021)
- [3D Neural Scene Representations for Visuomotor Control](https://mlsanity.com/paper/2107.04004/) (2021)
- [The Limits and Potentials of Deep Learning for Robotics](https://mlsanity.com/paper/1804.06557/) (2018)
- [NeRF-Supervision: Learning Dense Object Descriptors from Neural Radiance Fields](https://mlsanity.com/paper/2203.01913/) (2022)
- [Stochastic Neural Radiance Fields: Quantifying Uncertainty in Implicit 3D Representations](https://mlsanity.com/paper/2109.02123/) (2021)
- [Conditional-Flow NeRF: Accurate 3D Modelling with Reliable Uncertainty Quantification](https://mlsanity.com/paper/2203.10192/) (2022)
- [Semantics for Robotic Mapping, Perception and Interaction: A Survey](https://mlsanity.com/paper/2101.00443/) (2021)
- [Neural Volume Rendering: NeRF And Beyond](https://mlsanity.com/paper/2101.05204/) (2020)
- [Reinforcement Learning with Neural Radiance Fields](https://mlsanity.com/paper/2206.01634/) (2022)
- [Advances in Inference and Representation for Simultaneous Localization and Mapping](https://mlsanity.com/paper/2103.05041/) (2021)

## Built on by (most cited first, 30 of 60)

- [PUP 3D-GS: Principled Uncertainty Pruning for 3D Gaussian Splatting](https://mlsanity.com/paper/2406.10219/) (2024)
- [Bayes' Rays: Uncertainty Quantification for Neural Radiance Fields](https://mlsanity.com/paper/2309.03185/) (2023)
- [FisherRF: Active View Selection and Uncertainty Quantification for Radiance Fields using Fisher Information](https://mlsanity.com/paper/2311.17874/) (2023)
- [CountGD: Multi-Modal Open-World Counting](https://mlsanity.com/paper/2407.04619/) (2024)
- [Deceptive-NeRF/3DGS: Diffusion-Generated Pseudo-Observations for High-Quality Sparse-View Reconstruction](https://mlsanity.com/paper/2305.15171/) (2023)
- [Active Neural Mapping](https://mlsanity.com/paper/2308.16246/) (2023)
- [4D Gaussian Splatting in the Wild with Uncertainty-Aware Regularization](https://mlsanity.com/paper/2411.08879/) (2024)
- [Neural Visibility Field for Uncertainty-Driven Active Mapping](https://mlsanity.com/paper/2406.06948/) (2024)
- [Modeling uncertainty for Gaussian Splatting](https://mlsanity.com/paper/2403.18476/) (2024)
- [Estimating 3D Uncertainty Field: Quantifying Uncertainty for Neural Radiance Fields](https://mlsanity.com/paper/2311.01815/) (2023)
- [NeRF Director: Revisiting View Selection in Neural Volume Rendering](https://mlsanity.com/paper/2406.08839/) (2024)
- [Free360: Layered Gaussian Splatting for Unbounded 360-Degree View Synthesis from Extremely Sparse and Unposed Views](https://mlsanity.com/paper/2503.24382/) (2025)
- [Sources of Uncertainty in 3D Scene Reconstruction](https://mlsanity.com/paper/2409.06407/) (2024)
- [Bayesian NeRF: Quantifying Uncertainty with Volume Density for Neural Implicit Fields](https://mlsanity.com/paper/2404.06727/) (2024)
- [Multimodal LLM Guided Exploration and Active Mapping using Fisher Information](https://mlsanity.com/paper/2410.17422/) (2024)
- [Regularized Multi-Decoder Ensemble for an Error-Aware Scene Representation Network](https://mlsanity.com/paper/2407.19082/) (2024)
- [These Magic Moments: Differentiable Uncertainty Quantification of Radiance Field Models](https://mlsanity.com/paper/2503.14665/) (2025)
- [PRIMU: Uncertainty Estimation for Novel Views in Gaussian Splatting from Primitive-Based Representations of Error and Coverage](https://mlsanity.com/paper/2508.02443/) (2025)
- [ExploreGS: Explorable 3D Scene Reconstruction with Virtual Camera Samplings and Diffusion Priors](https://mlsanity.com/paper/2508.06014/) (2025)
- [Uncertainty Matters in Dynamic Gaussian Splatting for Monocular 4D Reconstruction](https://mlsanity.com/paper/2510.12768/) (2025)
- [View-Dependent Uncertainty Estimation of 3D Gaussian Splatting](https://mlsanity.com/paper/2504.07370/) (2025)
- [Peering into the Unknown: Active View Selection with Neural Uncertainty Maps for 3D Reconstruction](https://mlsanity.com/paper/2506.14856/) (2025)
- [Predictive Photometric Uncertainty in Gaussian Splatting for Novel View Synthesis](https://mlsanity.com/paper/2603.22786/) (2026)
- [Uncertainty-driven 3D Gaussian Splatting Active Mapping via Anisotropic Visibility Field](https://mlsanity.com/paper/2605.30342/) (2026)
- [Exploiting Priors from 3D Diffusion Models for RGB-Based One-Shot View Planning](https://mlsanity.com/paper/2403.16803/) (2024)
- [Uncertainty-aware Active Learning of NeRF-based Object Models for Robot Manipulators using Visual and Re-orientation Actions](https://mlsanity.com/paper/2404.01812/) (2024)
- [Auto3R: Automated 3D Reconstruction and Scanning via Data-driven Uncertainty Quantification](https://mlsanity.com/paper/2512.04528/) (2025)
- [Evidential Neural Radiance Fields](https://mlsanity.com/paper/2602.23574/) (2026)
- [Indoor Scene Reconstruction with Fine-Grained Details Using Hybrid Representation and Normal Prior Enhancement](https://mlsanity.com/paper/2309.07640/) (2023)
- [Instant Uncertainty Calibration of NeRFs Using a Meta-Calibrator](https://mlsanity.com/paper/2312.02350/) (2023)

## Abstract (arXiv)

We show that ensembling effectively quantifies model uncertainty in Neural Radiance Fields (NeRFs) if a density-aware epistemic uncertainty term is considered. The naive ensembles investigated in prior work simply average rendered RGB images to quantify the model uncertainty caused by conflicting explanations of the observed scene. In contrast, we additionally consider the termination probabilities along individual rays to identify epistemic model uncertainty due to a lack of knowledge about the parts of a scene unobserved during training. We achieve new state-of-the-art performance across established uncertainty quantification benchmarks for NeRFs, outperforming methods that require complex changes to the NeRF architecture and training regime. We furthermore demonstrate that NeRF uncertainty can be utilised for next-best view selection and model refinement.