HybridFlow: Quantification of Aleatoric and Epistemic Uncertainty with a Single Hybrid Model

TL;DR

HybridFlow introduces a modular two-stage framework that unifies aleatoric and epistemic uncertainty modeling by using a conditional normalizing flow to learn $p(y|x)$ and a separate probabilistic predictor that consumes the flow’s latent representation to quantify model uncertainty. This decoupled design preserves predictive accuracy while delivering calibrated, sharp uncertainty estimates across diverse regression tasks, including depth estimation, standard UCI benchmarks, and a scientific ice sheet emulator. By enabling seamless integration with existing predictors and avoiding joint loss calibration challenges, HybridFlow offers a practical, flexible approach to robust uncertainty quantification with broad applicability in scientific and engineering domains.

Abstract

Uncertainty quantification is critical for ensuring robustness in high-stakes machine learning applications. We introduce HybridFlow, a modular hybrid architecture that unifies the modeling of aleatoric and epistemic uncertainty by combining a Conditional Masked Autoregressive normalizing flow for estimating aleatoric uncertainty with a flexible probabilistic predictor for epistemic uncertainty. The framework supports integration with any probabilistic model class, allowing users to easily adapt HybridFlow to existing architectures without sacrificing predictive performance. HybridFlow improves upon previous uncertainty quantification frameworks across a range of regression tasks, such as depth estimation, a collection of regression benchmarks, and a scientific case study of ice sheet emulation. We also provide empirical results of the quantified uncertainty, showing that the uncertainty quantified by HybridFlow is calibrated and better aligns with model error than existing methods for quantifying aleatoric and epistemic uncertainty. HybridFlow addresses a key challenge in Bayesian deep learning, unifying aleatoric and epistemic uncertainty modeling in a single robust framework.

Figures (4)

• Figure 1: HybridFlow architecture, with depth estimation with the NYU Depth v2 dataset as the example task. A Conditional Masked Autoregressive Flow (CMAF) learns the density of input data (x) through a latent representation (z). The latent representation and the input data are concatenated and used as inputs to a probabilistic predictor model. From the predictor model, the prediction and the epistemic uncertainty can be estimated. The aleatoric uncertainty is estimated by calculating the variance of samples generated from the learned data distribution by the CMAF.
• Figure 2: Reconstruction results from the autoencoder trained on the NYU Depth v2 dataset. Each column shows the true depth image (left) and the corresponding reconstructed depth map (right). The autoencoder effectively compresses high-dimensional input data into a 256-dimensional latent representation, enabling efficient dimensionality reduction while preserving fine-grained spatial details.
• Figure 3: Visualization of depth estimation results using the HybridFlow framework compared to the BNLL seitzer2022pitfalls and NLL kendall2017uncertainties on the NYU Depth v2 dataset. Each row showcases a sample scene, with the columns representing (from left to right): the input RGB image, ground truth depth map, predicted depth map, model error, aleatoric uncertainty map, and epistemic uncertainty map. The predicted depth maps closely align with the ground truth, demonstrating accurate scene reconstruction. The aleatoric uncertainty maps highlight areas with inherent measurement noise, such as changes in image lighting (glare) or occlusion boundaries, while the epistemic uncertainty maps identify regions of the scene where the predictive model is uncertain. These visualizations illustrate the HybridFlow model's ability to provide both accurate predictions and reliable uncertainty quantification.
• Figure 4: Scatter plots of aleatoric (x-axis) vs. epistemic (y-axis) uncertainty across test points for three datasets and three methods.