# On the Expressiveness of Approximate Inference in Bayesian Neural Networks

arXiv: https://arxiv.org/abs/1909.00719 · HTML: https://mlsanity.com/paper/1909.00719/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2019.

## Built on by (most cited first, 5 of 5)

- [A Primer on Bayesian Neural Networks: Review and Debates](https://mlsanity.com/paper/2309.16314/) (2023)
- [Density Uncertainty Layers for Reliable Uncertainty Estimation](https://mlsanity.com/paper/2306.12497/) (2023)
- [PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantification](https://mlsanity.com/paper/2310.06923/) (2023)
- [Understanding Uncertainty in Bayesian Deep Learning](https://mlsanity.com/paper/2106.13055/) (2021)
- [Applications of Gaussian Processes at Extreme Lengthscales: From Molecules to Black Holes](https://mlsanity.com/paper/2303.14291/) (2023)

## Abstract (arXiv)

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational methods in approximating the Bayesian predictive distribution. For single-hidden layer ReLU BNNs, we prove a fundamental limitation in function-space of two of the most commonly used distributions defined in weight-space: mean-field Gaussian and Monte Carlo dropout. We find there are simple cases where neither method can have substantially increased uncertainty in between well-separated regions of low uncertainty. We provide strong empirical evidence that exact inference does not have this pathology, hence it is due to the approximation and not the model. In contrast, for deep networks, we prove a universality result showing that there exist approximate posteriors in the above classes which provide flexible uncertainty estimates. However, we find empirically that pathologies of a similar form as in the single-hidden layer case can persist when performing variational inference in deeper networks. Our results motivate careful consideration of the implications of approximate inference methods in BNNs.