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Evaluating and Calibrating Uncertainty Prediction in Regression Tasks
Dan Levi, Liran Gispan, Niv Giladi, Ethan Fetaya
TL;DR
Regression applications need uncertainty estimates that are informative, especially for safety-critical decisions, but existing interval-based calibration can treat independent uncertainty as perfectly calibrated. The paper proposes an MSE-based calibration definition, histogram-style evaluation, and simple variance scaling, and reports improved calibration in real-world applications.
Problem
Existing regression-uncertainty calibration has severe limitations in distinguishing informative from non-informative predictions.
Method
The paper defines calibration by matching predicted variance with empirical mean square error, evaluates it by grouping similar uncertainty predictions, and rescales variance using a separate recalibration set.
Results
The proposed method shows good calibration on a real-world dataset using a simple parametric model that scales uncertainty by a constant factor.
Takeaways & Limitations
The new definition avoids declaring uncertainty perfectly calibrated when it is uncorrelated with actual uncertainty, while simple scaling improves calibration in real-world applications.
Takeaways & Limitations
The analytic counterexample uses a Cauchy distribution for simplicity, although the argument does not depend on that distribution.
Abstract
from arXiv · showhide
Predicting not only the target but also an accurate measure of uncertainty is important for many machine learning applications and in particular safety-critical ones. In this work we study the calibration of uncertainty prediction for regression tasks which often arise in real-world systems. We show that the existing definition for calibration of a regression uncertainty [Kuleshov et al. 2018] has severe limitations in distinguishing informative from non-informative uncertainty predictions. We propose a new definition that escapes this caveat and an evaluation method using a simple histogram-based approach. Our method clusters examples with similar uncertainty prediction and compares the prediction with the empirical uncertainty on these examples. We also propose a simple, scaling-based calibration method that preforms as well as much more complex ones. We show results on both a synthetic, controlled problem and on the object detection bounding-box regression task using the COCO and KITTI datasets.
1 Introduction
The paper motivates calibrated regression uncertainty for applications where predictions alone are insufficient, identifies flaws in existing calibration, and proposes a simpler definition and scaling method.
- Motivation: Safety-critical systems need predictions accompanied by confidence information to support decisions.Examples include self-driving agents and Kalman-filter tracking, which uses location-estimation variance.
- Uncertainty estimation: Regression uncertainty can be produced by outputting a Gaussian mean and a separately predicted standard deviation.The paper contrasts this direct approach with Bayesian neural networks and ensembles.
- Problem: Existing interval-based calibration averages across the dataset and can calibrate practically any output distribution, including uninformative uncertainty.The paper reports both empirical and analytical evidence for this limitation.
- Contributions: The paper proposes a calibration definition closer to classification, replacing misclassification probability with mean square error.It also introduces an ECE-like evaluation metric and rescales predicted Gaussian variance by minimizing NLL on a separate recalibration set.
- Contributions: A simple scaling method is reported to reduce calibration error as well as more complex methods on large real-world vision datasets.The method is compared with approaches including histogram binning, Bayesian binning, isotonic regression, and temperature scaling.
2 Confidence-intervals based calibration
This section reviews interval-based regression calibration and demonstrates that it can report perfect calibration even when predicted uncertainty is independent of actual uncertainty.
- Interval-based method: The reviewed interval-based method evaluates regression calibration through predicted confidence intervals and includes recalibration and reliability-diagram diagnostics.Its practical procedure uses a recalibration dataset, empirical CDF values, and isotonic regression.
- Notation: A regression forecaster outputs a distribution for each input, represented in continuous tasks by a probability density and its corresponding CDF.The notation distinguishes the input-target distribution from each prediction-specific distribution.
- Interval-based definition: The interval definition requires predicted CDF values to fall below percentile p for exactly p of the data.The paper presents this as equivalent to matching the predicted and empirical CDFs asymptotically.
- Limitation: The methodology can achieve perfect calibration even when output CDFs are statistically independent of empirical uncertainty.Because the relevant mapping is non-decreasing, isotonic regression can fit it perfectly up to sampling error.
- Analytic example: An analytic example uses a normal target and a Cauchy prediction with a random independent scale, yet the interval criterion is satisfied exactly.The ratio of independent normal variables yields the Cauchy construction used to establish the result.
- Analytic example: Although the Cauchy distribution simplifies exposition, the argument does not depend on it and can be replaced by an implicitly constructed Gaussian distribution.The paper notes that the Gaussian construction is less familiar and explicit.
3 Our method
The method defines regression uncertainty calibration by matching predicted variance to empirical mean squared error at each uncertainty level, then evaluates this relationship with binned diagnostics. It supplements calibration with uncertainty-dispersion measures and uses likelihood-based standard-deviation scaling for recalibration.
- Calibration definition: Calibration requires the expected mean squared error to match the predicted variance for each uncertainty value.The definition replaces classification misclassification probability with regression mean squared error.
- Calibration definition: The proposed definition avoids averaging together examples with different predicted variances, unlike the prior credible-interval formulation.Practical evaluation still requires binning, but the definition separates uncertainty levels conceptually.
- Evaluation: The reliability diagram plots empirical RMSE against root mean variance per uncertainty bin, with calibrated predictions near the identity line.Additional scores summarize calibration error across bins.
- Evaluation: Expected Normalized Calibration Error averages per-bin calibration gaps after normalizing by each bin’s mean predicted variance.The normalization accounts for naturally larger errors at larger predicted variances and parallels classification ECE.
- Evaluation: The coefficient of variation complements ENCE because a homogeneous uncertainty prediction can achieve zero ENCE while remaining uninformative per example.The method treats ENCE as the primary calibration measure and CV as a secondary dispersion diagnostic.
- Recalibration: Standard-deviation scaling recalibrates Gaussian predictions by multiplying each predicted STD by one factor while leaving the predicted mean unchanged.The factor is optimized using recalibration-set NLL, with diagnosis performed on a separate validation set; scaling performed on par with percentile-based isotonic regression.
4 Experimental results
Experiments evaluate the proposed uncertainty calibration approach on synthetic regression and object-detection bounding-box regression. The method exposes random uncertainty, improves calibration of predicted uncertainty, and achieves comparable performance to the interval-based method with simpler computation.
- The experiments cover a controlled synthetic regression problem and bounding-box regression, including trained and random uncertainty predictions.
- For KITTI bounding-box regression, reliability diagrams compare empirical RMSE with RMV before and after calibration across the four positional outputs.
- 50,000 synthetic samples are generated with target standard deviation x_t, while random per-example uncertainty is drawn from Uniform[1, 10].
- The synthetic experiment shows interval calibration can make random uncertainty appear almost perfectly calibrated under its own definition, although it remains uncorrelated with actual uncertainty.
- Learned scaling factors between 1.1 and 1.2 reduce ENCE, while cv remains unchanged because uniform STD scaling leaves it invariant.
- Both the proposed and interval-based methods considerably improve ENCE with comparable performance, while the proposed calibration uses only a scalar parameter and no numerical integration.
5 Conclusions
The paper identifies serious flaws in existing regression calibration and proposes a new definition, evaluation metrics, and simple recalibration method. The method shows significant improvement in real-world applications.
- The paper exposes serious flaws in the existing approach to defining and evaluating calibration for regression uncertainty.
- It proposes a new regression-calibration definition, evaluation metrics, and a simple recalibration method that shows significant improvement in real-world applications.
A.1 Synthetic regression problem
Additional synthetic experiments distinguish random uncertainty from uncertainty learned by the network. The proposed evaluation and calibration behavior differs sharply between these settings.
- Before calibration, randomly assigned uncertainty is entirely uncorrelated with true uncertainty, and recalibration leaves that lack of correlation unchanged.
- The proposed reliability diagrams and ENCE measures reveal the random uncertainty’s poor calibration before and after recalibration.
- In the trained experiment, the network almost perfectly learns the correct uncertainty because the synthetic problem is simple and data availability is high.
- Both the proposed and interval methods have little effect when uncertainty is already correctly predicted, while the proposed approach distinguishes this case from random uncertainty.
A.2 Bounding box regression for object detection
Bounding-box experiments add an uncertainty branch to an R-FCN detector and evaluate calibration for four positional regression outputs. Random or untrained uncertainty remains non-informative under the proposed metrics after calibration.
- The R-FCN regression branch predicts four bounding-box parameters, (t_x, t_y, t_w, t_h), treated as separate regression outputs.
- The added uncertainty branch predicts log variances for corresponding Gaussian outputs, while the original regression outputs represent Gaussian means.
- COCO trains the detector, while separate KITTI subsets are used for uncertainty calibration and validation to reduce overfitting risk.
- Untrained uncertainty is randomly initialized and therefore uncorrelated with true uncertainty; reliability diagrams show highly non-calibrated uncertainty under the proposed metrics before and after calibration.
- Figures compare synthetic pre- and post-calibration STD relationships and reliability diagrams, including interval-based calibration comparisons.