# Fast Predictive Uncertainty for Classification with Bayesian Deep Networks

arXiv: https://arxiv.org/abs/2003.01227 · HTML: https://mlsanity.com/paper/2003.01227/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Mar 2020.

## Built on by (most cited first, 30 of 31)

- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [Prior and Posterior Networks: A Survey on Evidential Deep Learning Methods For Uncertainty Estimation](https://mlsanity.com/paper/2110.03051/) (2021)
- [Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI](https://mlsanity.com/paper/2402.00809/) (2024)
- [A Simple Approach to Improve Single-Model Deep Uncertainty via Distance-Awareness](https://mlsanity.com/paper/2205.00403/) (2022)
- [RegMixup: Mixup as a Regularizer Can Surprisingly Improve Accuracy and Out Distribution Robustness](https://mlsanity.com/paper/2206.14502/) (2022)
- [CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks](https://mlsanity.com/paper/2401.05043/) (2024)
- [Low-rank extended Kalman filtering for online learning of neural networks from streaming data](https://mlsanity.com/paper/2305.19535/) (2023)
- [Random-Set Neural Networks (RS-NN)](https://mlsanity.com/paper/2307.05772/) (2023)
- [Active Learning in Bayesian Neural Networks with Balanced Entropy Learning Principle](https://mlsanity.com/paper/2105.14559/) (2021)
- [Posterior Refinement Improves Sample Efficiency in Bayesian Neural Networks](https://mlsanity.com/paper/2205.10041/) (2022)
- [Credal Wrapper of Model Averaging for Uncertainty Estimation in Classification](https://mlsanity.com/paper/2405.15047/) (2024)
- [A Unified Evaluation Framework for Epistemic Predictions](https://mlsanity.com/paper/2501.16912/) (2025)
- [Deep Classifiers with Label Noise Modeling and Distance Awareness](https://mlsanity.com/paper/2110.02609/) (2021)
- [Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'](https://mlsanity.com/paper/2505.04950/) (2025)
- [Conformal Prediction as Bayesian Quadrature](https://mlsanity.com/paper/2502.13228/) (2025)
- [Analytic Mutual Information in Bayesian Neural Networks](https://mlsanity.com/paper/2201.09815/) (2022)
- [Enhancing Transfer Learning with Flexible Nonparametric Posterior Sampling](https://mlsanity.com/paper/2403.07282/) (2024)
- [laplax -- Laplace Approximations with JAX](https://mlsanity.com/paper/2507.17013/) (2025)
- [Uncertainty-Aware Autonomous Vehicles: Predicting the Road Ahead](https://mlsanity.com/paper/2510.22680/) (2025)
- [Laplace Matching for fast Approximate Inference in Latent Gaussian Models](https://mlsanity.com/paper/2105.03109/) (2021)
- [Deep Gaussian Mixture Ensembles](https://mlsanity.com/paper/2306.07235/) (2023)
- [EventSleep: Sleep Activity Recognition with Event Cameras](https://mlsanity.com/paper/2404.01801/) (2024)
- [LEDiT: Your Length-Extrapolatable Diffusion Transformer without Positional Encoding](https://mlsanity.com/paper/2503.04344/) (2025)
- [Rethinking Approximate Gaussian Inference in Classification](https://mlsanity.com/paper/2502.03366/) (2025)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)
- [Function-Space Regularization for Deep Bayesian Classification](https://mlsanity.com/paper/2307.06055/) (2023)
- [Uncertainty-driven Embedding Convolution](https://mlsanity.com/paper/2507.20718/) (2025)
- [Laplace-Bridged Randomized Smoothing for Fast Certified Robustness](https://mlsanity.com/paper/2604.24993/) (2026)
- [Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification](https://mlsanity.com/paper/2605.12208/) (2026)

## Abstract (arXiv)

In Bayesian Deep Learning, distributions over the output of classification neural networks are often approximated by first constructing a Gaussian distribution over the weights, then sampling from it to receive a distribution over the softmax outputs. This is costly. We reconsider old work (Laplace Bridge) to construct a Dirichlet approximation of this softmax output distribution, which yields an analytic map between Gaussian distributions in logit space and Dirichlet distributions (the conjugate prior to the Categorical distribution) in the output space. Importantly, the vanilla Laplace Bridge comes with certain limitations. We analyze those and suggest a simple solution that compares favorably to other commonly used estimates of the softmax-Gaussian integral. We demonstrate that the resulting Dirichlet distribution has multiple advantages, in particular, more efficient computation of the uncertainty estimate and scaling to large datasets and networks like ImageNet and DenseNet. We further demonstrate the usefulness of this Dirichlet approximation by using it to construct a lightweight uncertainty-aware output ranking for ImageNet.