Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Density-Informed Pseudo-Counts for Calibrated Evidential Deep Learning

Pietro Carlotti, Nevena Gligić, Arya Farahi

TL;DR

This work reframes Evidential Deep Learning (EDL) as amortized variational inference in a hierarchical Bayesian model with a tempered pseudo-likelihood, revealing that standard EDL conflates epistemic and aleatoric uncertainty and overconfidently assigns high certainty to OOD data. It introduces Density-Informed Pseudo-count EDL (DIP-EDL), which decouples class prediction from uncertainty by combining a density estimator of the covariate distribution with a discriminative network for $P^*_{Y|X}$, yielding distribution-aware uncertainty that concentrates in high-density regions and contracts toward a uniform prior for OOD inputs. The authors prove asymptotic concentration for DIP-EDL and empirically show improved in-distribution calibration and out-of-distribution detection on MNIST and CIFAR-10 against several baselines. This work provides a principled, modular uncertainty quantification framework with strong theoretical backing and practical robustness to distributional shift.

Abstract

Evidential Deep Learning (EDL) is a popular framework for uncertainty-aware classification that models predictive uncertainty via Dirichlet distributions parameterized by neural networks. Despite its popularity, its theoretical foundations and behavior under distributional shift remain poorly understood. In this work, we provide a principled statistical interpretation by proving that EDL training corresponds to amortized variational inference in a hierarchical Bayesian model with a tempered pseudo-likelihood. This perspective reveals a major drawback: standard EDL conflates epistemic and aleatoric uncertainty, leading to systematic overconfidence on out-of-distribution (OOD) inputs. To address this, we introduce Density-Informed Pseudo-count EDL (DIP-EDL), a new parametrization that decouples class prediction from the magnitude of uncertainty by separately estimating the conditional label distribution and the marginal covariate density. This separation preserves evidence in high-density regions while shrinking predictions toward a uniform prior for OOD data. Theoretically, we prove that DIP-EDL achieves asymptotic concentration. Empirically, we show that our method enhances interpretability and improves robustness and uncertainty calibration under distributional shift.

Density-Informed Pseudo-Counts for Calibrated Evidential Deep Learning

TL;DR

This work reframes Evidential Deep Learning (EDL) as amortized variational inference in a hierarchical Bayesian model with a tempered pseudo-likelihood, revealing that standard EDL conflates epistemic and aleatoric uncertainty and overconfidently assigns high certainty to OOD data. It introduces Density-Informed Pseudo-count EDL (DIP-EDL), which decouples class prediction from uncertainty by combining a density estimator of the covariate distribution with a discriminative network for , yielding distribution-aware uncertainty that concentrates in high-density regions and contracts toward a uniform prior for OOD inputs. The authors prove asymptotic concentration for DIP-EDL and empirically show improved in-distribution calibration and out-of-distribution detection on MNIST and CIFAR-10 against several baselines. This work provides a principled, modular uncertainty quantification framework with strong theoretical backing and practical robustness to distributional shift.

Abstract

Evidential Deep Learning (EDL) is a popular framework for uncertainty-aware classification that models predictive uncertainty via Dirichlet distributions parameterized by neural networks. Despite its popularity, its theoretical foundations and behavior under distributional shift remain poorly understood. In this work, we provide a principled statistical interpretation by proving that EDL training corresponds to amortized variational inference in a hierarchical Bayesian model with a tempered pseudo-likelihood. This perspective reveals a major drawback: standard EDL conflates epistemic and aleatoric uncertainty, leading to systematic overconfidence on out-of-distribution (OOD) inputs. To address this, we introduce Density-Informed Pseudo-count EDL (DIP-EDL), a new parametrization that decouples class prediction from the magnitude of uncertainty by separately estimating the conditional label distribution and the marginal covariate density. This separation preserves evidence in high-density regions while shrinking predictions toward a uniform prior for OOD data. Theoretically, we prove that DIP-EDL achieves asymptotic concentration. Empirically, we show that our method enhances interpretability and improves robustness and uncertainty calibration under distributional shift.
Paper Structure (39 sections, 14 theorems, 48 equations, 1 figure, 9 tables)

This paper contains 39 sections, 14 theorems, 48 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Statistical Interpretation of EDL
  3. Problem Setup and Notation
  4. Amortized Variational Inference
  5. Empirical Risk Minimization
  6. Choice of the Temperature Parameter
  7. Experiments
  8. Experimental Configuration
  9. Datasets.
  10. Baselines.
  11. Implementation Details.
  12. Performance Metrics.
  13. Experiments with MNIST as ID dataset
  14. Experiments with CIFAR-10 as ID dataset
  15. Ablation Study
  16. ...and 24 more sections

Key Result

Proposition 2.2

Suppose model in Definition def:independent_categorical_dirichlet_model, the posterior distribution of each vector of class probabilities is $p_{i}\!\mid\!Y_{1:n} \sim \pi_{Y_{i}}\!=\!\mathrm{Dir}(\alpha+e_{Y_{i}})$ where $e_{Y_{i}}$ denotes the one-hot encoding of $Y_{i}$. Furthermore, the joint po

Figures (1)

  • Figure 1: Confidence Distributions for DIP-EDL.Left: DIP-EDL demonstrates a near-perfect performance on MNIST where OOD samples collapse to the uniform prior (0.1). Right: On CIFAR-10, DIP-EDL shows a high-confidence tail for OOD samples, while remaining confident on ID data. This demonstrates the density estimator's ability to differentiate between the two but struggle to capture all the details of complex, high-dimensional datasets, and explains the higher Brier Score despite good AUROC results.

Theorems & Definitions (28)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Theorem 2.10
  • ...and 18 more