# Bayes' Rays: Uncertainty Quantification for Neural Radiance Fields

arXiv: https://arxiv.org/abs/2309.03185 · HTML: https://mlsanity.com/paper/2309.03185/

## Key facts

- Cited by 91 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 60 references that are in the map.
- First posted on arXiv 6 Sep 2023.

## Builds on (most cited first, 30 of 36)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [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)
- [Mip-NeRF 360: Unbounded Anti-Aliased Neural Radiance Fields](https://mlsanity.com/paper/2111.12077/) (2021)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [NeRF++: Analyzing and Improving Neural Radiance Fields](https://mlsanity.com/paper/2010.07492/) (2020)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Depth-supervised NeRF: Fewer Views and Faster Training for Free](https://mlsanity.com/paper/2107.02791/) (2021)
- [RegNeRF: Regularizing Neural Radiance Fields for View Synthesis from Sparse Inputs](https://mlsanity.com/paper/2112.00724/) (2021)
- [Nerfstudio: A Modular Framework for Neural Radiance Field Development](https://mlsanity.com/paper/2302.04264/) (2023)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [NerfingMVS: Guided Optimization of Neural Radiance Fields for Indoor Multi-view Stereo](https://mlsanity.com/paper/2109.01129/) (2021)
- [NEAT: Neural Attention Fields for End-to-End Autonomous Driving](https://mlsanity.com/paper/2109.04456/) (2021)
- [Optimizing the Latent Space of Generative Networks](https://mlsanity.com/paper/1707.05776/) (2017)
- [Uncertainty Estimates and Multi-Hypotheses Networks for Optical Flow](https://mlsanity.com/paper/1802.07095/) (2018)
- [ActiveNeRF: Learning where to See with Uncertainty Estimation](https://mlsanity.com/paper/2209.08546/) (2022)
- [Estimating and Exploiting the Aleatoric Uncertainty in Surface Normal Estimation](https://mlsanity.com/paper/2109.09881/) (2021)
- [Stochastic Segmentation Networks: Modelling Spatially Correlated Aleatoric Uncertainty](https://mlsanity.com/paper/2006.06015/) (2020)
- [RobustNeRF: Ignoring Distractors with Robust Losses](https://mlsanity.com/paper/2302.00833/) (2023)
- [LOLNeRF: Learn from One Look](https://mlsanity.com/paper/2111.09996/) (2021)
- [Stochastic Neural Radiance Fields: Quantifying Uncertainty in Implicit 3D Representations](https://mlsanity.com/paper/2109.02123/) (2021)
- [Nerfbusters: Removing Ghostly Artifacts from Casually Captured NeRFs](https://mlsanity.com/paper/2304.10532/) (2023)
- [Conditional-Flow NeRF: Accurate 3D Modelling with Reliable Uncertainty Quantification](https://mlsanity.com/paper/2203.10192/) (2022)
- [Density-aware NeRF Ensembles: Quantifying Predictive Uncertainty in Neural Radiance Fields](https://mlsanity.com/paper/2209.08718/) (2022)
- [Neural Ensemble Search for Uncertainty Estimation and Dataset Shift](https://mlsanity.com/paper/2006.08573/) (2020)
- [NeurAR: Neural Uncertainty for Autonomous 3D Reconstruction with Implicit Neural Representations](https://mlsanity.com/paper/2207.10985/) (2022)
- [CamP: Camera Preconditioning for Neural Radiance Fields](https://mlsanity.com/paper/2308.10902/) (2023)

## Built on by (most cited first, 30 of 91)

- [3D Gaussian Splatting as Markov Chain Monte Carlo](https://mlsanity.com/paper/2404.09591/) (2024)
- [PUP 3D-GS: Principled Uncertainty Pruning for 3D Gaussian Splatting](https://mlsanity.com/paper/2406.10219/) (2024)
- [NeRF On-the-go: Exploiting Uncertainty for Distractor-free NeRFs in the Wild](https://mlsanity.com/paper/2405.18715/) (2024)
- [FisherRF: Active View Selection and Uncertainty Quantification for Radiance Fields using Fisher Information](https://mlsanity.com/paper/2311.17874/) (2023)
- [NeRFiller: Completing Scenes via Generative 3D Inpainting](https://mlsanity.com/paper/2312.04560/) (2023)
- [Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI](https://mlsanity.com/paper/2402.00809/) (2024)
- [SpotlessSplats: Ignoring Distractors in 3D Gaussian Splatting](https://mlsanity.com/paper/2406.20055/) (2024)
- [NARUTO: Neural Active Reconstruction from Uncertain Target Observations](https://mlsanity.com/paper/2402.18771/) (2024)
- [FlowR: Flowing from Sparse to Dense 3D Reconstructions](https://mlsanity.com/paper/2504.01647/) (2025)
- [Drone-assisted Road Gaussian Splatting with Cross-view Uncertainty](https://mlsanity.com/paper/2408.15242/) (2024)
- [Modeling uncertainty for Gaussian Splatting](https://mlsanity.com/paper/2403.18476/) (2024)
- [POp-GS: Next Best View in 3D-Gaussian Splatting with P-Optimality](https://mlsanity.com/paper/2503.07819/) (2025)
- [GauSS-MI: Gaussian Splatting Shannon Mutual Information for Active 3D Reconstruction](https://mlsanity.com/paper/2504.21067/) (2025)
- [Accelerating Neural Field Training via Soft Mining](https://mlsanity.com/paper/2312.00075/) (2023)
- [AONeuS: A Neural Rendering Framework for Acoustic-Optical Sensor Fusion](https://mlsanity.com/paper/2402.03309/) (2024)
- [The Role of Predictive Uncertainty and Diversity in Embodied AI and Robot Learning](https://mlsanity.com/paper/2405.03164/) (2024)
- [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)
- [Active Neural Mapping at Scale](https://mlsanity.com/paper/2409.20276/) (2024)
- [Distractor-free Generalizable 3D Gaussian Splatting](https://mlsanity.com/paper/2411.17605/) (2024)
- [NeRF-US: Removing Ultrasound Imaging Artifacts from Neural Radiance Fields in the Wild](https://mlsanity.com/paper/2408.10258/) (2024)
- [PhysGS: Bayesian-Inferred Gaussian Splatting for Physical Property Estimation](https://mlsanity.com/paper/2511.18570/) (2025)
- [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)
- [SO-NeRF: Active View Planning for NeRF using Surrogate Objectives](https://mlsanity.com/paper/2312.03266/) (2023)
- [Uncertainty-Aware Normal-Guided Gaussian Splatting for Surface Reconstruction from Sparse Image Sequences](https://mlsanity.com/paper/2503.11172/) (2025)
- [View-Dependent Uncertainty Estimation of 3D Gaussian Splatting](https://mlsanity.com/paper/2504.07370/) (2025)
- [Predictive Photometric Uncertainty in Gaussian Splatting for Novel View Synthesis](https://mlsanity.com/paper/2603.22786/) (2026)

## Abstract (arXiv)

Neural Radiance Fields (NeRFs) have shown promise in applications like view synthesis and depth estimation, but learning from multiview images faces inherent uncertainties. Current methods to quantify them are either heuristic or computationally demanding. We introduce BayesRays, a post-hoc framework to evaluate uncertainty in any pre-trained NeRF without modifying the training process. Our method establishes a volumetric uncertainty field using spatial perturbations and a Bayesian Laplace approximation. We derive our algorithm statistically and show its superior performance in key metrics and applications. Additional results available at: https://bayesrays.github.io.