# Quantifying Uncertainty in Discrete-Continuous and Skewed Data with Bayesian Deep Learning

arXiv: https://arxiv.org/abs/1802.04742 · HTML: https://mlsanity.com/paper/1802.04742/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2018.

## Built on by (most cited first, 13 of 13)

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- [Deep Ensembles to Improve Uncertainty Quantification of Statistical Downscaling Models under Climate Change Conditions](https://mlsanity.com/paper/2305.00975/) (2023)
- [Variational Bayesian Pseudo-Coreset](https://mlsanity.com/paper/2502.21143/) (2025)
- [The Sensitivity of Variational Bayesian Neural Network Performance to Hyperparameters](https://mlsanity.com/paper/2509.20574/) (2025)
- [Multi-Quantile Regression for Extreme Precipitation Downscaling](https://mlsanity.com/paper/2605.12762/) (2026)

## Abstract (arXiv)

Deep Learning (DL) methods have been transforming computer vision with innovative adaptations to other domains including climate change. For DL to pervade Science and Engineering (S&E) applications where risk management is a core component, well-characterized uncertainty estimates must accompany predictions. However, S&E observations and model-simulations often follow heavily skewed distributions and are not well modeled with DL approaches, since they usually optimize a Gaussian, or Euclidean, likelihood loss. Recent developments in Bayesian Deep Learning (BDL), which attempts to capture uncertainties from noisy observations, aleatoric, and from unknown model parameters, epistemic, provide us a foundation. Here we present a discrete-continuous BDL model with Gaussian and lognormal likelihoods for uncertainty quantification (UQ). We demonstrate the approach by developing UQ estimates on `DeepSD', a super-resolution based DL model for Statistical Downscaling (SD) in climate applied to precipitation, which follows an extremely skewed distribution. We find that the discrete-continuous models outperform a basic Gaussian distribution in terms of predictive accuracy and uncertainty calibration. Furthermore, we find that the lognormal distribution, which can handle skewed distributions, produces quality uncertainty estimates at the extremes. Such results may be important across S&E, as well as other domains such as finance and economics, where extremes are often of significant interest. Furthermore, to our knowledge, this is the first UQ model in SD where both aleatoric and epistemic uncertainties are characterized.