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Uncertainty Quantification for Deep Regression using Contextualised Normalizing Flows
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Proposes MCNF, a post hoc uncertainty quantification method that conditions a neural spline normalizing flow on Monte Carlo Dropout prior samples and summary statistics to produce full predictive distributions and prediction intervals for pretrained regression models.
Problem
Deep regression models need calibrated uncertainty estimates that capture distributional information (multimodality, asymmetry) without costly Bayesian methods or retraining the base model.
Approach
Model the prediction error (y minus MCD prior sample) with a context-conditioned neural spline normalizing flow whose context is the MCD sample mean, log-variance, and a hidden representation of the input; train the flow post hoc with a temperature-weighted likelihood that downweights outliers.
Key results
- MCNF achieves marginal coverage close to the 90% target on most datasets (e.g., 0.920 on Concrete, 0.927 on Protein) while producing narrower median interval sizes than CQR, MCCP and MCQR (e.g., 0.491 vs 0.660 for CQR on Concrete).
- MCNF attains the lowest median MAE on most datasets, e.g., 0.085 on Concrete vs 0.355 for CQR, and 0.202 on Protein vs 0.935 for CQR.
- MCD, which lacks aleatoric calibration, reaches only 0.601 coverage on Concrete and 0.354 on Protein.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the MCNF normalizing-flow correction to MCD | DQR uncertainty estimates · Protein | median MAE: 0.245±0.010 -> 0.202±0.009 | Helped |
| Add the MCNF normalizing-flow correction to MCD | DQR uncertainty estimates · Boston Housing | median interval size: 0.254±0.023 -> 0.409±0.038 | Mixed |
| Add MCD prediction-error resampling to propagate epistemic uncertainty (ablation) | MCNF · Protein | median MAE: 0.186±0.008 -> 0.202±0.009 | Helped |
| Add MCD prediction-error resampling to propagate epistemic uncertainty (ablation) | MCNF · Romano-Original | median MAE: 0.496±0.154 -> 0.406±0.206 | Helped |
| Vary MCNF training epochs, NF sample count, or MCD sample count among tested values (ablation) | MCNF · Boston Housing, Abalone, and Concrete | coverage | No clear effect |
| Increase temperature parameter τ (ablation) | MCNF outlier weighting · Synthetic outlier datasets | coverage | Mixed |
| Use a well-trained rather than underfitted base predictive model | MCNF · Concrete, Superconductivity, and Protein | interval size | Helped |
| Use MCNF rather than MCD | DQR uncertainty estimates · Romano-Mod | coverage: 0.540±0.0513 -> 0.952±0.015 | Helped |
| Use MCNF rather than NF without epistemic-uncertainty propagation (ablation) | DQR uncertainty estimates · Romano-Original | median MAE: 0.496±0.154 -> 0.406±0.206 | Helped |
Surprising findings
- MCNF's MAE is an order of magnitude lower than conformal baselines, which the authors attribute to CP's homogeneous interval adjustment being suited only to near-Gaussian distributions.
- Removing epistemic propagation (the NF-only variant) gives consistently narrower intervals and lower coverage than full MCNF.
Limitations
- Requires MCD sampling at inference (n_MCD samples per input), adding computational overhead.
- Outlier-weighting temperature tau has no universal optimum and likely needs tuning on a calibration set; MCNF is sensitive to outliers when effect size is near zero.
Experimental setup
Datasets: Boston Housing Concrete Abalone Tertiary Protein Structure Wave energy Superconductivity Solubility Romano-Original Romano-Mod
Benchmarks: Boston Housing Concrete Abalone Protein Wave Superconductivity Solubility Romano-Original Romano-Mod
Metrics: marginal coverage median interval size median MAE quantile MAE
Baselines: CQR MCCP MCD MCQR DQR
Builds on: Monte Carlo Dropout Conformalized Quantile Regression Monte Carlo Conformal Prediction Neural Spline Flows Normalizing Flows
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