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Revisiting the Evaluation of Uncertainty Estimation and Its Application to Explore Model Complexity-Uncertainty Trade-Off

Yukun Ding, Jinglan Liu, Jinjun Xiong, Yiyu Shi

arXiv:1903.02050v3cs.LGstat.ML

TL;DR

The effect of model complexity on uncertainty-estimation quality has received little study, despite the importance of understanding performance trade-offs. This paper examines selective prediction and confidence calibration, proposes metrics to address issues in existing evaluations, and finds that their complexity-uncertainty patterns differ.

  • Problem

    The effect of model complexity on uncertainty-estimation quality is almost unknown, although understanding this trade-off matters when choosing models under resource constraints.

    Source: §1 · p. 2

  • Method

    The paper analyzes issues in common metrics for selective prediction and confidence calibration, proposes AURC and adaptive binning, and uses the new metrics to study model complexity.

    Source: §1 · p. 2

  • Results

    Uncertainty-estimation quality changes significantly with model complexity for selective prediction but is insensitive to complexity for confidence calibration.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The study reports that selective prediction and confidence calibration have different complexity-uncertainty trade-offs.

    Source: §6 · p. 8

    Estimating accuracy within confidence bins requires enough samples, limiting accuracy estimation when bins are sparsely populated.

    Source: §4.2 · p. 5

Abstract

from arXiv · show

Accurately estimating uncertainties in neural network predictions is of great importance in building trusted DNNs-based models, and there is an increasing interest in providing accurate uncertainty estimation on many tasks, such as security cameras and autonomous driving vehicles. In this paper, we focus on the two main use cases of uncertainty estimation, i.e. selective prediction and confidence calibration. We first reveal potential issues of commonly used quality metrics for uncertainty estimation in both use cases, and propose our new metrics to mitigate them. We then apply these new metrics to explore the trade-off between model complexity and uncertainty estimation quality, a critically missing work in the literature. Our empirical experiment results validate the superiority of the proposed metrics, and some interesting trends about the complexity-uncertainty trade-off are observed.

1. Introduction

The paper examines selective prediction and confidence calibration, finding shortcomings in common uncertainty-estimation metrics and proposing alternatives. It uses those metrics to study how model complexity relates to uncertainty quality.

  • The paper focuses on selective prediction and confidence calibration as two use cases of uncertainty estimation.Selective prediction uses confidence to separate correct from incorrect predictions, while calibration aims for confidence to approximate the probability of correctness.Source: §1 · p. 1
  • For selective prediction, AURC is reliable when prediction models change and consistent with AUROC and AUPR when they stay the same.The paper proposes AURC as the primary metric for this use case.Source: §1 · p. 2
  • For confidence calibration, adaptive binning is proposed for ECE and MCE after basic binning is found to miss some high-confidence calibration errors.The paper reports that the new binning strategy performs better empirically.Source: §1 · p. 2
  • Uncertainty-estimation quality changes significantly with model complexity for selective prediction but is insensitive to complexity for confidence calibration.The study uses its proposed metrics to explore this trade-off and validate their effectiveness and robustness.Source: §1 · p. 2
  • The paper identifies potential problems with existing metrics and proposes alternatives intended to provide more reliable and informative evaluations.It also presents an exploration of the complexity-uncertainty trade-off.Source: §1 · p. 2

2. Related Works

Prior work uses several approaches to estimate uncertainty and commonly evaluates selective prediction with AUROC and AUPR, and calibration with ECE and MCE. The paper situates its work against these methods and their limitations.

  • Uncertainty estimates may come from softmax probability, Monte Carlo dropout, or learned confidence estimation, and may be integrated during training or applied afterward.The paper's analysis does not assume a particular source of confidence scores or a correlation between confidence and prediction.Source: §2 · p. 2
  • AUROC and AUPR are common selective-prediction metrics, but E-AURC also fails to account for accuracy differences when prediction models vary.This matters when comparing models, training-altering algorithms, or uncertainty methods such as MC-dropout and ensembles.Source: §2 · p. 2
  • ECE and MCE depend heavily on binning, and equal-size bins may remain too inflexible for highly non-uniform confidence distributions.The paper contrasts this with its adaptive binning method; Brier Score and NLL are described as favoring prediction quality over calibration by design.Source: §2 · p. 2

3. Problem Setting

The paper frames uncertainty evaluation around predicted labels, confidence scores, and correctness. It defines selective prediction through abstention and calibration through the gap between confidence and empirical correctness, with ECE and MCE computed using bins.

  • For each input, the model produces a predicted label and confidence score; correctness is encoded as 1 for a correct prediction and 0 otherwise.These quantities define the distribution over confidence and correctness used in the paper's setting.Source: §3 · p. 3
  • Selective prediction thresholds confidence, abstaining on low-confidence inputs while retaining predictions above the threshold.Ideally, the abstained set contains wrong predictions and the retained set contains correct predictions.Source: §3 · p. 3
  • Confidence calibration aims for a score in [0, 1] that reflects the probability that the prediction is correct.Calibration error is the difference between confidence and empirical correctness at that confidence.Source: §3 · p. 3
  • In practice, ECE and MCE partition the confidence range into bins and compare average accuracy with average confidence within each bin.The standard practice described uses 10 equal-range bins.Source: §3 · p. 3
  • ECE and MCE are proper but not strictly proper scoring rules, and even strictly proper rules do not guarantee reliable evaluation and comparison.A reliability diagram plots empirical accuracy and calibration error by bin and visualizes how ECE and MCE are calculated.Source: §3 · p. 3

4. Evaluation Metrics: Issues & Solutions

The section argues that common uncertainty metrics can mislead when prediction models differ and can hide calibration errors because of binning. It presents AURC for selective prediction and adaptive binning metrics for confidence calibration.

4.1. Selective Prediction

  • AUROC and AUPR can rank larger-m DenseNet variants better despite their worse selective-prediction quality, including after a 0.2% accuracy change.The example varies m by changing the most uncertain predictions, with larger m indicating worse selective-prediction quality.Source: §4.1 · pp. 3–4, Figure 1 · p. 4
  • AURC gives the correct selective-prediction ranking when underlying models differ, whereas AUROC and AUPR can misleadingly favor worse-performing DenseNet-m variants.The reported comparison says AUROC and AUPR favor larger m, contrary to prior knowledge; AURC is the only reliable evaluation in this setting.Source: §4.1 · p. 4, Table 1 · p. 4
  • For equal-accuracy models, RC-curve dominance matches ROC-curve dominance, and AURC extends to task-specific risks such as 1 − Dice for segmentation.The section also presents RC curves as displaying cost-performance trade-offs that are not visible in ROC or PR curves.Source: §4.1 · pp. 4–5

4.2. Confidence Calibration

  • ECE, MCE, and Reliability Diagrams can hide large calibration errors: accuracy is only 50% for confidence [0.9, 0.91], although the [0.9, 1] calibration error is 0.0215.The high error over [0.9, 0.96] can be obscured by averaging, since most samples have confidence in [0.98, 1.0].Source: §4.2 · p. 5, Figure 2 · p. 5
  • Calibration estimates also face internal compensation within bins and inaccurate accuracy estimates in small bins, while equal-size bins can create overly wide sparse-region bins.More samples can address inaccurate accuracy estimates, but not undetectable error and internal compensation caused by a highly nonuniform confidence distribution.Source: §4.2 · pp. 5–6
  • Adaptive binning balances bin range against sample count, and its AECE and AMCE metrics capture calibration error while alleviating the described problems.The reported implementation uses an 80% confidence interval, has O(|D|) complexity, and achieves the best results on the example network among the shown binning strategies.Source: §4.2 · p. 6, Figure 3 · p. 6

5. Effect of Model Complexity on Uncertainty Estimation

The section compares selective prediction and confidence calibration, then evaluates how model complexity affects each. It finds that selective-prediction quality rises with model size, while calibration quality remains nearly flat.

5.1. Relation Between the Two Use Cases

  • Selective prediction depends on the ranking of confidence scores, whereas calibration depends on their specific values, so the two qualities need not correlate.With limited learning capability, the model should be optimized for the intended use case.Source: §5.1 · p. 7
  • Perfect uncertainty estimates for both use cases require correctly identifying each prediction’s correctness, which the authors describe as almost impossible in practice.The paper states this condition as an if-and-only-if result.Source: §5.1 · p. 7

5.2. Experiment Results

  • The experiments use maximum softmax probability and temperature scaling with DenseNet and WideResNet on Cifar10 and Cifar100, plus U-Nets of different widths for heart segmentation.The classification datasets cover different levels of task difficulty and accuracy.Source: §5.2 · pp. 7–8
  • Adaptive binning produces an average of 12.6 bins on Cifar10 and 21.2 on Cifar100; AECE is generally slightly higher than ECE.Cifar100’s flatter confidence distribution permits more bins with accurate accuracy estimation.Source: §5.2 · pp. 7–8
  • AURC shows selective-prediction performance increasing with model size, while AUPR decreases because it omits full-coverage accuracy.This contrast is shown in Figure 4.Source: §5.2 · p. 7, Figure 4 · p. 8
  • MCE can substantially exceed AMCE for DenseNet and becomes unstable for both networks on Cifar10, where confidence distributions are more non-uniform.The authors attribute the discrepancies to inaccurate accuracy estimates in bins with few samples.Source: §5.2 · p. 8
  • AECE and AMCE remain almost flat as model complexity changes, showing little sensitivity to model size for confidence calibration.The observed pattern combines model accuracy, temperature-scaling effectiveness, and confidence distribution.Source: §5.2 · p. 7, Figure 5 · p. 8
  • For U-Net heart segmentation, AUPR is not meaningful because the networks are optimized for Dice, while MCE remains unstable.ECE and AECE are almost equal in this small dataset, where overfitting and easy background voxels concentrate predictions near confidence 1.Source: §5.2 · p. 8

6. Conclusions

The paper concludes that its new metrics address issues in existing uncertainty-estimation evaluations and that selective prediction and confidence calibration have different complexity-uncertainty trade-offs.

6. Conclusions

  • The authors conclude that selective prediction and confidence calibration have different complexity-uncertainty trade-offs, as shown by applying the proposed metrics.They present the metrics as justified alternatives to existing measures that may produce unreliable or misleading results.Source: §6 · p. 8

A. Examples of Adaptive Binning

Figure 6 presents reliability diagrams for various models; the text reports that temperature scaling reduces calibration error and adaptive binning uses more bins for a harder calibrated dataset.

A. Examples of Adaptive Binning

B Experiment Results of Medical Image Segmentation

The section reports experiment results on the Multi-Modality Whole Heart Segmentation dataset, as shown in Figure 7.

B Experiment Results of Medical Image Segmentation

C.1 Proof of Theorem 1

Theorem 2 states that, for networks with equal accuracy, dominance in ROC space is equivalent to dominance in Risk-Coverage space, regardless of how each network’s uncertainty is measured.

C.1 Proof of Theorem 1

  • For any two networks with the same accuracy, ROC-space dominance is equivalent to Risk-Coverage-space dominance, even when their uncertainty methods differ.Source: §C.1 · p. 10
  • The proof uses contradiction, treating wrong predictions as positive samples and correct predictions as negative samples; equal accuracy gives equal counts of each.The proof defines TN, TP, FN, FP, TPR, and FPR as the corresponding confusion-matrix quantities and rates.Source: §C.1 · p. 10
  • At equal coverage, the assumed lower risk for network A implies fewer false negatives and more true negatives than network B.Source: §C.1 · p. 11
  • With equal numbers of positive and negative samples, those inequalities imply A has higher TPR and lower FPR than B, contradicting B’s ROC dominance.Source: §C.1 · p. 11

C.2 Proof of Proposition 1

Proposition 3 gives a condition for estimated ECE to equal ground-truth ECE under any bin selection; otherwise, estimated ECE is smaller.

C.2 Proof of Proposition 1

  • For any bin selection, estimated ECE equals ground-truth ECE only when each bin’s conditional expected confidence stays entirely at or above, or entirely at or below, its range value.Otherwise, estimated ECE is strictly smaller than ground-truth ECE.Source: §C.2 · p. 11
  • The proof assumes enough samples and a solvable conditional expected-confidence value so that ground-truth ECE is meaningful.Source: §C.2 · p. 11
  • The proof partitions samples into bins by their distinct confidence values, without requiring a uniform bin partition.Source: §C.2 · p. 11
  • The equality condition is that within each bin, conditional expected confidence values are all on the same side of the bin’s range value.Source: §C.2 · p. 11

C.3 Proof of Proposition 2

Proposition 4 characterizes perfect uncertainty estimation for selective prediction and confidence calibration: the uncertainty values must be binary, with conditional mean correctness matching each value. The proof establishes sufficiency and necessity of these conditions.

C.3 Proof of Proposition 2

  • Perfect uncertainty estimation for selective prediction and confidence calibration holds if and only if r is binary and E[c|r=0]=0 and E[c|r=1]=1.The proposition states the necessary and sufficient conditions on samples and their conditional expected correctness.Source: §C.3 · p. 11
  • Under these conditions, expected calibration error is zero and every correct sample receives greater uncertainty than every incorrect sample.The proof derives both properties directly from the binary values of r and the endpoint conditional means.Source: §C.3 · p. 11
  • Conversely, zero expected calibration error implies E[c|r]=r; any intermediate r would give correct and incorrect samples equal uncertainty, violating selective ranking.Therefore r must be binary, and the conditional means at r=0 and r=1 are respectively 0 and 1.Source: §C.3 · p. 11
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