# Improved uncertainty quantification for neural networks with Bayesian last layer

arXiv: https://arxiv.org/abs/2302.10975 · HTML: https://mlsanity.com/paper/2302.10975/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 21 Feb 2023.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Learning in High Dimension Always Amounts to Extrapolation](https://mlsanity.com/paper/2110.09485/) (2021)
- [Efficient representation and approximation of model predictive control laws via deep learning](https://mlsanity.com/paper/1806.10644/) (2018)
- [Benchmarking the Neural Linear Model for Regression](https://mlsanity.com/paper/1912.08416/) (2019)
- [Deep Latent-Variable Kernel Learning](https://mlsanity.com/paper/2005.08467/) (2020)

## Built on by (most cited first, 12 of 12)

- [Variational Bayesian Last Layers](https://mlsanity.com/paper/2404.11599/) (2024)
- [Uncertainty Quantification and Propagation in Surrogate-based Bayesian Inference](https://mlsanity.com/paper/2312.05153/) (2023)
- [Bayesian deep learning for cosmic volumes with modified gravity](https://mlsanity.com/paper/2309.00612/) (2023)
- [Generative vs. Discriminative modeling under the lens of uncertainty quantification](https://mlsanity.com/paper/2406.09172/) (2024)
- [Achieving Well-Informed Decision-Making in Drug Discovery: A Comprehensive Calibration Study using Neural Network-Based Structure-Activity Models](https://mlsanity.com/paper/2407.14185/) (2024)
- [Is the Last Layer Sufficient for Uncertainty Quantification?](https://mlsanity.com/paper/2605.30741/) (2026)
- [Multivariate Bayesian Last Layer for Regression with Uncertainty Quantification and Decomposition](https://mlsanity.com/paper/2405.01761/) (2024)
- [Bayesian Surrogate Training on Multiple Data Sources: A Hybrid Modeling Strategy](https://mlsanity.com/paper/2412.11875/) (2024)
- [Variational Bayesian Pseudo-Coreset](https://mlsanity.com/paper/2502.21143/) (2025)
- [Scalable Deep Basis Kernel Gaussian Processes](https://mlsanity.com/paper/2505.18526/) (2025)
- [Efficient Bayesian Deep Ensembles via Analytic Predictive Inference](https://mlsanity.com/paper/2607.06776/) (2026)
- [Variational Filtering for Bayesian Last Layers with an Independent Binary Probit Surrogate](https://mlsanity.com/paper/2610.16578/) (2026)

## Abstract (arXiv)

Uncertainty quantification is an important task in machine learning - a task in which standardneural networks (NNs) have traditionally not excelled. This can be a limitation for safety-critical applications, where uncertainty-aware methods like Gaussian processes or Bayesian linear regression are often preferred. Bayesian neural networks are an approach to address this limitation. They assume probability distributions for all parameters and yield distributed predictions. However, training and inference are typically intractable and approximations must be employed. A promising approximation is NNs with Bayesian last layer (BLL). They assume distributed weights only in the linear output layer and yield a normally distributed prediction. To approximate the intractable Bayesian neural network, point estimates of the distributed weights in all but the last layer should be obtained by maximizing the marginal likelihood. This has previously been challenging, as the marginal likelihood is expensive to evaluate in this setting. We present a reformulation of the log-marginal likelihood of a NN with BLL which allows for efficient training using backpropagation. Furthermore, we address the challenge of uncertainty quantification for extrapolation points. We provide a metric to quantify the degree of extrapolation and derive a method to improve the uncertainty quantification for these points. Our methods are derived for the multivariate case and demonstrated in a simulation study. In comparison to Bayesian linear regression with fixed features, and a Bayesian neural network trained with variational inference, our proposed method achieves the highest log-predictive density on test data.