[Paper Review] Towards Principled Uncertainty Estimation for Deep Neural Networks

Richard Harang, Ethan M. Rudd|arXiv (Cornell University)|Oct 29, 2018

Gaussian Processes and Bayesian Inference21 references4 citations

TL;DR

This paper proposes a unified hierarchical deep neural network that jointly estimates three types of uncertainty—model capacity, intrinsic data, and open set—by combining Bayesian inference, invertible normalizing flows, and discriminative classification. The model enables efficient, end-to-end per-sample uncertainty estimation with minimal computational overhead and provably bounds open-set risk under Gaussian latent space assumptions.

ABSTRACT

When the cost of misclassifying a sample is high, it is useful to have an accurate estimate of uncertainty in the prediction for that sample. There are also multiple types of uncertainty which are best estimated in different ways, for example, uncertainty that is intrinsic to the training set may be well-handled by a Bayesian approach, while uncertainty introduced by shifts between training and query distributions may be better-addressed by density/support estimation. In this paper, we examine three types of uncertainty: model capacity uncertainty, intrinsic data uncertainty, and open set uncertainty, and review techniques that have been derived to address each one. We then introduce a unified hierarchical model, which combines methods from Bayesian inference, invertible latent density inference, and discriminative classification in a single end-to-end deep neural network topology to yield efficient per-sample uncertainty estimation in a detection context. This approach addresses all three uncertainty types and can readily accommodate prior/base rates for binary detection. We then discuss how to extend this model to a more generic multiclass recognition context.

Motivation & Objective

• To address the limitations of existing uncertainty estimation methods that fail to distinguish between distinct types of uncertainty in deep neural networks.
• To develop a scalable, end-to-end deep learning architecture that simultaneously models model capacity, intrinsic data, and open set uncertainty.
• To provide a principled, calibrated uncertainty estimate per sample that supports practical deployment in high-stakes applications.
• To incorporate prior/base rates naturally into the model for binary detection tasks.
• To extend the framework to multiclass recognition under a multinomial noise model with Dirichlet priors.

Proposed method

• The model integrates a Bayesian-inspired probabilistic head with a normalizing flow-based density estimator in a shared deep neural network architecture.
• It uses invertible latent density inference to model the likelihood of inputs in a disentangled latent space, enabling uncertainty quantification via density estimation.
• A discriminative cross-entropy head ensures accurate classification while the latent space is regularized to separate class-specific distributions.
• The model is trained end-to-end using a joint objective combining classification loss and likelihood maximization for uncertainty estimation.
• Class-specific Gaussian outputs in the latent space are used to model in-class support, reducing the risk of high-density regions between clusters.
• The framework naturally accommodates prior probabilities and base rates for binary detection, with theoretical bounds on open-set risk under convergence to Gaussianity.

Experimental results

Research questions

• RQ1 How can a single deep learning model effectively estimate multiple distinct types of uncertainty—model capacity, intrinsic data, and open set—simultaneously?
• RQ2 Can a unified architecture combining Bayesian inference, normalizing flows, and discriminative learning yield calibrated uncertainty estimates with low inference cost?
• RQ3 How does the model’s uncertainty estimation perform under distributional shifts or out-of-distribution inputs?
• RQ4 What is the impact of using class-specific latent Gaussians versus a single shared Gaussian on density estimation and uncertainty calibration?
• RQ5 How can prior knowledge and base rates be incorporated into the uncertainty estimation process for binary detection tasks?

Key findings

• The model achieves efficient per-sample uncertainty estimation with only one forward pass, making it suitable for real-time applications.
• Empirical evaluation on a character recognition task showed that 17% of false positives were due to 'l' being misclassified as '1', and 12% were due to 'I' being misclassified as '1'.
• The model distinguishes plausible mistakes (e.g., 'O' as '0', 'S' as '5') from clear errors, improving assessment of density estimator quality.
• The use of class-specific latent Gaussians mitigates the issue of spurious high-density regions between clusters, a known limitation of single-Gaussian flow models.
• The framework provably bounds open-set risk under the assumption that the latent density converges to a Gaussian, providing theoretical grounding for uncertainty calibration.
• The model extends naturally to multiclass recognition under a multinomial noise model with Dirichlet conjugate priors, though formal uncertainty quantification in this regime remains a direction for future work.