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Quantifying Deep Learning Model Uncertainty in Conformal Prediction

Hamed Karimi, Reza Samavi

TL;DR

The paper addresses reliable predictive uncertainty in high-stakes settings like medical AI by leveraging Conformal Prediction (CP), which provides prediction sets $\mathcal{C}(x,\widehat{q})$ with marginal coverage guarantees $P(y \in \mathcal{C}) \ge 1-\delta$ via calibration data and a score $s(x,y)$. It introduces a probabilistic uncertainty-quantification framework for CP, defining conformal model uncertainty $U_{\mathcal{C}}(x_{val})$ with bounds $\mathcal{L}_{\mathcal{C}}$ and $\mathcal{H}_{\mathcal{C}}$ based on the prediction-set size $m$, number of classes $K$, calibration size $n$, and coverage error $\delta$, and a pure-model uncertainty $\widehat{u}_{\mathcal{C}} = \frac{m+\delta-1}{K}$. A special case of $m=0$ yields maximal uncertainty, while larger $m$ and larger $\delta$ increase the bound width and hence the quantified uncertainty, captured by $d_{\mathcal{C}} = \frac{1+\widehat{u}_{\mathcal{C}}}{n+1}$. The approach enables comparisons with Bayesian and Evidential UQ methods and provides a principled, certified framework for uncertainty in CP-based classification, with practical implications for medical decision-making; it also discusses related CP variants APS and RAPS that improve adaptivity and reduce set size while maintaining coverage.

Abstract

Precise estimation of predictive uncertainty in deep neural networks is a critical requirement for reliable decision-making in machine learning and statistical modeling, particularly in the context of medical AI. Conformal Prediction (CP) has emerged as a promising framework for representing the model uncertainty by providing well-calibrated confidence levels for individual predictions. However, the quantification of model uncertainty in conformal prediction remains an active research area, yet to be fully addressed. In this paper, we explore state-of-the-art CP methodologies and their theoretical foundations. We propose a probabilistic approach in quantifying the model uncertainty derived from the produced prediction sets in conformal prediction and provide certified boundaries for the computed uncertainty. By doing so, we allow model uncertainty measured by CP to be compared by other uncertainty quantification methods such as Bayesian (e.g., MC-Dropout and DeepEnsemble) and Evidential approaches.

Quantifying Deep Learning Model Uncertainty in Conformal Prediction

TL;DR

The paper addresses reliable predictive uncertainty in high-stakes settings like medical AI by leveraging Conformal Prediction (CP), which provides prediction sets with marginal coverage guarantees via calibration data and a score . It introduces a probabilistic uncertainty-quantification framework for CP, defining conformal model uncertainty with bounds and based on the prediction-set size , number of classes , calibration size , and coverage error , and a pure-model uncertainty . A special case of yields maximal uncertainty, while larger and larger increase the bound width and hence the quantified uncertainty, captured by . The approach enables comparisons with Bayesian and Evidential UQ methods and provides a principled, certified framework for uncertainty in CP-based classification, with practical implications for medical decision-making; it also discusses related CP variants APS and RAPS that improve adaptivity and reduce set size while maintaining coverage.

Abstract

Precise estimation of predictive uncertainty in deep neural networks is a critical requirement for reliable decision-making in machine learning and statistical modeling, particularly in the context of medical AI. Conformal Prediction (CP) has emerged as a promising framework for representing the model uncertainty by providing well-calibrated confidence levels for individual predictions. However, the quantification of model uncertainty in conformal prediction remains an active research area, yet to be fully addressed. In this paper, we explore state-of-the-art CP methodologies and their theoretical foundations. We propose a probabilistic approach in quantifying the model uncertainty derived from the produced prediction sets in conformal prediction and provide certified boundaries for the computed uncertainty. By doing so, we allow model uncertainty measured by CP to be compared by other uncertainty quantification methods such as Bayesian (e.g., MC-Dropout and DeepEnsemble) and Evidential approaches.
Paper Structure (6 sections, 2 theorems, 14 equations, 2 figures)

This paper contains 6 sections, 2 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on Conformal Prediction
  3. Related Work
  4. Uncertainty Quantification in Prediction Sets
  5. Interpretation of the Uncertainty Quantification
  6. Conclusion

Key Result

Theorem 1

Consider $\{(x_i,y_i) \in \mathcal{X}_{cal}\}_{i=1}^n$ and $(x_{val},y_{val}) \in \mathcal{X}_{val}$ are i.i.d. and unseen data as $n$ calibration data points and a validation data point, respectively. Let $\delta$ be the user-chosen coverage error level, $\widehat{q}$ is the $1-\delta$ quantile of

Figures (2)

  • Figure 1: The conformal model uncertainty associated with prediction sets of different sizes $m$ based on the variation of error level $\delta$
  • Figure 2: The conformal model uncertainty variation $d_{\mathcal{C}}$ associated with prediction sets of different sizes $m$ based on the variation of calibration data size $n$ when $\delta$ is fixed

Theorems & Definitions (3)

  • Theorem 1: Conformal Coverage Guarantee cpinductive_cp
  • Theorem 2: Conformal Uncertainty Quantification
  • proof