# Learnable Uncertainty under Laplace Approximations

arXiv: https://arxiv.org/abs/2010.02720 · HTML: https://mlsanity.com/paper/2010.02720/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2020.

## Built on by (most cited first, 29 of 29)

- [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)
- [Decomposing Uncertainty for Large Language Models through Input Clarification Ensembling](https://mlsanity.com/paper/2311.08718/) (2023)
- [A Survey on Uncertainty Quantification Methods for Deep Learning](https://mlsanity.com/paper/2302.13425/) (2023)
- [Variational Bayesian Last Layers](https://mlsanity.com/paper/2404.11599/) (2024)
- [Do Bayesian Neural Networks Need To Be Fully Stochastic?](https://mlsanity.com/paper/2211.06291/) (2022)
- [Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI](https://mlsanity.com/paper/2402.00809/) (2024)
- [Uncertainty Estimation by Fisher Information-based Evidential Deep Learning](https://mlsanity.com/paper/2303.02045/) (2023)
- [A Simple Approach to Improve Single-Model Deep Uncertainty via Distance-Awareness](https://mlsanity.com/paper/2205.00403/) (2022)
- [Uncertainty quantification in fine-tuned LLMs using LoRA ensembles](https://mlsanity.com/paper/2402.12264/) (2024)
- [Post-hoc Uncertainty Learning using a Dirichlet Meta-Model](https://mlsanity.com/paper/2212.07359/) (2022)
- [Are Uncertainty Quantification Capabilities of Evidential Deep Learning a Mirage?](https://mlsanity.com/paper/2402.06160/) (2024)
- [Being a Bit Frequentist Improves Bayesian Neural Networks](https://mlsanity.com/paper/2106.10065/) (2021)
- [Dropout Injection at Test Time for Post Hoc Uncertainty Quantification in Neural Networks](https://mlsanity.com/paper/2302.02924/) (2023)
- [Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization](https://mlsanity.com/paper/2304.08309/) (2023)
- [laplax -- Laplace Approximations with JAX](https://mlsanity.com/paper/2507.17013/) (2025)
- [Adversarial Attacks Against Uncertainty Quantification](https://mlsanity.com/paper/2309.10586/) (2023)
- [Ranking over Regression for Bayesian Optimization and Molecule Selection](https://mlsanity.com/paper/2410.09290/) (2024)
- [Modern Neural Networks for Small Tabular Datasets: The New Default for Field-Scale Digital Soil Mapping?](https://mlsanity.com/paper/2508.09888/) (2025)
- [From predictions to confidence intervals: an empirical study of conformal prediction methods for in-context learning](https://mlsanity.com/paper/2504.15722/) (2025)
- [CLAPS: Aleatoric-Epistemic Scaling via Last-Layer Laplace for Conformal Regression](https://mlsanity.com/paper/2512.01384/) (2025)
- [Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogates](https://mlsanity.com/paper/2603.11052/) (2026)
- [Bayesian Domain Invariant Learning via Posterior Generalization of Parameter Distributions](https://mlsanity.com/paper/2310.16277/) (2023)
- [Connecting Parameter Magnitudes and Hessian Eigenspaces at Scale using Sketched Methods](https://mlsanity.com/paper/2504.14701/) (2025)
- [On the Laplace Approximation as Model Selection Criterion for Gaussian Processes](https://mlsanity.com/paper/2403.09215/) (2024)
- [UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows](https://mlsanity.com/paper/2503.23236/) (2025)
- [Monte Carlo Functional Regularisation for Continual Learning](https://mlsanity.com/paper/2508.13006/) (2025)
- [Kalman Bayesian Transformer](https://mlsanity.com/paper/2509.10695/) (2025)
- [Physics-Informed Neural Networks for Methane Sorption: Cross-Gas Transfer Learning, Ensemble Collapse Under Physics Constraints, and Monte Carlo Dropout Uncertainty Quantification](https://mlsanity.com/paper/2604.13992/) (2026)

## Abstract (arXiv)

Laplace approximations are classic, computationally lightweight means for constructing Bayesian neural networks (BNNs). As in other approximate BNNs, one cannot necessarily expect the induced predictive uncertainty to be calibrated. Here we develop a formalism to explicitly "train" the uncertainty in a decoupled way to the prediction itself. To this end, we introduce uncertainty units for Laplace-approximated networks: Hidden units associated with a particular weight structure that can be added to any pre-trained, point-estimated network. Due to their weights, these units are inactive -- they do not affect the predictions. But their presence changes the geometry (in particular the Hessian) of the loss landscape, thereby affecting the network's uncertainty estimates under a Laplace approximation. We show that such units can be trained via an uncertainty-aware objective, improving standard Laplace approximations' performance in various uncertainty quantification tasks.