# These Magic Moments: Differentiable Uncertainty Quantification of Radiance Field Models

arXiv: https://arxiv.org/abs/2503.14665 · HTML: https://mlsanity.com/paper/2503.14665/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 52 references that are in the map.
- First posted on arXiv 18 Mar 2025.

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

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Mip-NeRF: A Multiscale Representation for Anti-Aliasing Neural Radiance Fields](https://mlsanity.com/paper/2103.13415/) (2021)
- [The Replica Dataset: A Digital Replica of Indoor Spaces](https://mlsanity.com/paper/1906.05797/) (2019)
- [Depth-supervised NeRF: Fewer Views and Faster Training for Free](https://mlsanity.com/paper/2107.02791/) (2021)
- [Language-driven Semantic Segmentation](https://mlsanity.com/paper/2201.03546/) (2022)
- [iMAP: Implicit Mapping and Positioning in Real-Time](https://mlsanity.com/paper/2103.12352/) (2021)
- [Gaussian Splatting SLAM](https://mlsanity.com/paper/2312.06741/) (2023)
- [LangSplat: 3D Language Gaussian Splatting](https://mlsanity.com/paper/2312.16084/) (2023)
- [Feature 3DGS: Supercharging 3D Gaussian Splatting to Enable Distilled Feature Fields](https://mlsanity.com/paper/2312.03203/) (2023)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [Urban Radiance Fields](https://mlsanity.com/paper/2111.14643/) (2021)
- [A Survey on 3D Gaussian Splatting](https://mlsanity.com/paper/2401.03890/) (2024)
- [3D Gaussian Splatting as Markov Chain Monte Carlo](https://mlsanity.com/paper/2404.09591/) (2024)
- [Depth-Regularized Optimization for 3D Gaussian Splatting in Few-Shot Images](https://mlsanity.com/paper/2311.13398/) (2023)
- [ActiveNeRF: Learning where to See with Uncertainty Estimation](https://mlsanity.com/paper/2209.08546/) (2022)
- [Bayes' Rays: Uncertainty Quantification for Neural Radiance Fields](https://mlsanity.com/paper/2309.03185/) (2023)
- [SGS-SLAM: Semantic Gaussian Splatting For Neural Dense SLAM](https://mlsanity.com/paper/2402.03246/) (2024)
- [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)
- [FisherRF: Active View Selection and Uncertainty Quantification for Radiance Fields using Fisher Information](https://mlsanity.com/paper/2311.17874/) (2023)
- [CG-SLAM: Efficient Dense RGB-D SLAM in a Consistent Uncertainty-aware 3D Gaussian Field](https://mlsanity.com/paper/2403.16095/) (2024)
- [Density-aware NeRF Ensembles: Quantifying Predictive Uncertainty in Neural Radiance Fields](https://mlsanity.com/paper/2209.08718/) (2022)
- [NeU-NBV: Next Best View Planning Using Uncertainty Estimation in Image-Based Neural Rendering](https://mlsanity.com/paper/2303.01284/) (2023)
- [The Unreasonable Effectiveness of Deep Evidential Regression](https://mlsanity.com/paper/2205.10060/) (2022)
- [Volume Rendering Digest (for NeRF)](https://mlsanity.com/paper/2209.02417/) (2022)
- [BeyondPixels: A Comprehensive Review of the Evolution of Neural Radiance Fields](https://mlsanity.com/paper/2306.03000/) (2023)
- [CATNIPS: Collision Avoidance Through Neural Implicit Probabilistic Scenes](https://mlsanity.com/paper/2302.12931/) (2023)
- [Modeling uncertainty for Gaussian Splatting](https://mlsanity.com/paper/2403.18476/) (2024)

## Built on by (most cited first, 6 of 6)

- [Confidence-Based Mesh Extraction from 3D Gaussians](https://mlsanity.com/paper/2603.24725/) (2026)
- [Next Best View Selections for Semantic and Dynamic 3D Gaussian Splatting](https://mlsanity.com/paper/2512.22771/) (2025)
- [ConfCtrl: Enabling Precise Camera Control in Video Diffusion via Confidence-Aware Interpolation](https://mlsanity.com/paper/2603.09819/) (2026)
- [GO-PRE: Goal-Oriented Next-Best-View Selection via Predictive Rendering Entropy for Active 3D Reconstruction](https://mlsanity.com/paper/2607.29037/) (2026)
- [A Survey of Spatial Memory Representations for Efficient Robot Navigation](https://mlsanity.com/paper/2604.16482/) (2026)
- [FillGS: Filling Observation Gaps in 4D Gaussian Splatting via Viewpoint-Time Selection and Generative Refinement](https://mlsanity.com/paper/2607.29284/) (2026)

## Abstract (arXiv)

This paper introduces a novel approach to uncertainty quantification for radiance fields by leveraging higher-order moments of the rendering equation. Uncertainty quantification is crucial for downstream tasks including view planning and scene understanding, where safety and robustness are paramount. However, the high dimensionality and complexity of radiance fields pose significant challenges for uncertainty quantification, limiting the use of these uncertainty quantification methods in high-speed decision-making. We demonstrate that the probabilistic nature of the rendering process enables efficient and differentiable computation of higher-order moments for radiance field outputs, including color, depth, and semantic predictions. Our method outperforms existing radiance field uncertainty estimation techniques while offering a more direct, computationally efficient, and differentiable formulation without the need for post-processing. Beyond uncertainty quantification, we also illustrate the utility of our approach in downstream applications such as next-best-view (NBV) selection and active ray sampling for neural radiance field training. Extensive experiments on synthetic and real-world scenes confirm the efficacy of our approach, which achieves state-of-the-art performance while maintaining simplicity.