Uncertainty Quantification for Deep Regression using Contextualised Normalizing Flows

TL;DR

MCNF introduces a post hoc uncertainty quantification framework for deep regression that pairs Monte Carlo Dropout priors with contextually conditioned normalizing flows to produce the full predictive distribution and calibrated prediction intervals. By summarizing the MCD posterior into a context vector and learning a flow over prediction errors, MCNF captures complex, including multimodal and heteroskedastic, uncertainty without retraining the base model. Across diverse benchmarks and a GNN setup, MCNF achieves competitive marginal coverage with smaller interval widths and lower MAE than state-of-the-art UQ methods, demonstrating practical value for safe decision-making. The approach also provides a configurable pathway for integrating with non-standard architectures and supports density evaluation, making it broadly applicable to regression tasks in high-stakes domains.

Abstract

Quantifying uncertainty in deep regression models is important both for understanding the confidence of the model and for safe decision-making in high-risk domains. Existing approaches that yield prediction intervals overlook distributional information, neglecting the effect of multimodal or asymmetric distributions on decision-making. Similarly, full or approximated Bayesian methods, while yielding the predictive posterior density, demand major modifications to the model architecture and retraining. We introduce MCNF, a novel post hoc uncertainty quantification method that produces both prediction intervals and the full conditioned predictive distribution. MCNF operates on top of the underlying trained predictive model; thus, no predictive model retraining is needed. We provide experimental evidence that the MCNF-based uncertainty estimate is well calibrated, is competitive with state-of-the-art uncertainty quantification methods, and provides richer information for downstream decision-making tasks.

Figures (8)

• Figure 1: Overview of the proposed MCNF UQ method for regression tasks. First, a set of samples is drawn using Monte Carlo Dropout (MCD). These samples are used to build a context vector that encodes the MCD predictive posterior and, ultimately, the input observation using a convenient set of summary statistics. These summary statistics are provided to the Normalizing Flow-based model as a context. Depending on the task, either a requested number of samples can be drawn from the Normalizing Flow by sampling the base distribution and submitting those samples through the forward pass, or the likelihood of the input observations can be assessed.
• Figure 2: MCNF predictions (y) against the synthetic Romano-Mod dataset, generated as described in the Appendix. Blue circles represent data observations within the 90% marginal coverage of MCNF, whereas the pink circles fall outside this range. The orange interval delineates the MCNF smoothed marginal coverage (superimposed over the unsmoothed interval, in gray). The broken green line represents the median marginal coverage. The ridge lines represent kernel density estimates of the predictive distributions for the MCNF samples (gray) and the prior MCD samples (black).
• Figure 3: (best viewed in color) Coverage and confidence interval results for a well-trained predictive model (dark colored) and an underfitted predictive model (light colored).
• Figure 4: Visualisation of the synthetic data produced based on Equation \ref{['eq:romanomod']}.
• Figure 5: Evaluation of MCNF on Boston Housing, Abalone, and Concrete datasets for different epoch values (20, 50, 100, and 150).