[Paper Review] Posterior Network: Uncertainty Estimation without OOD Samples via Density-Based Pseudo-Counts

Bertrand Charpentier, Daniel Zügner+1 more|arXiv (Cornell University)|Jun 16, 2020

Adversarial Robustness in Machine LearningComputer Science34 references60 citations

TL;DR

Posterior Network uses class-conditioned normalizing flows to build input-dependent Dirichlet posteriors, enabling uncertainty estimation without OOD data and improving OOD detection and calibration.

ABSTRACT

Accurate estimation of aleatoric and epistemic uncertainty is crucial to build safe and reliable systems. Traditional approaches, such as dropout and ensemble methods, estimate uncertainty by sampling probability predictions from different submodels, which leads to slow uncertainty estimation at inference time. Recent works address this drawback by directly predicting parameters of prior distributions over the probability predictions with a neural network. While this approach has demonstrated accurate uncertainty estimation, it requires defining arbitrary target parameters for in-distribution data and makes the unrealistic assumption that out-of-distribution (OOD) data is known at training time. In this work we propose the Posterior Network (PostNet), which uses Normalizing Flows to predict an individual closed-form posterior distribution over predicted probabilites for any input sample. The posterior distributions learned by PostNet accurately reflect uncertainty for in- and out-of-distribution data -- without requiring access to OOD data at training time. PostNet achieves state-of-the art results in OOD detection and in uncertainty calibration under dataset shifts.

Motivation & Objective

• Motivate reliable uncertainty estimation in neural networks for safety-critical domains.
• Address limitations of prior Dirichlet-based uncertainty models that require OOD data during training.
• Propose a framework that yields well-calibrated uncertainty for in-distribution and out-of-distribution data without access to OOD samples during training.
• Leverage normalizing flows to model an input-dependent posterior over class probabilities.

Proposed method

• Predict per-input Dirichlet parameters as a prior-plus-pseudo-count update beta^(i) = beta^(prior) + beta^(i) where alpha^(i)=beta^(prior)+beta^(i).
• Encode input to latent z via f_theta and learn class-conditional densities P(z|c;phi) with normalizing flows.
• Set beta^(prior) = 1 (flat prior) and derive beta^(i) from class-specific latent densities and ground-truth class counts.
• Train end-to-end using a Bayesian-like loss combining Uncertain Cross Entropy (UCE) and an entropy regularizer.
• Compute uncertainty from Dirichlet parameters; in-distribution data yield low epistemic uncertainty and near-true class posteriors, while OOD yields high epistemic uncertainty.
• Use density in latent space to enable OOD detection without needing explicit OOD samples.

Experimental results

Research questions

• RQ1 Can a model learn an input-dependent posterior over class probabilities without access to OOD data during training?
• RQ2 Do class-conditioned densities in latent space enable effective OOD detection and uncertainty calibration under distributional shifts?
• RQ3 How does PostNet balance aleatoric and epistemic uncertainty for ID versus OOD samples?
• RQ4 Is training with a Bayesian-like loss and density-based pseudo-counts more effective than traditional Prior Networks or ensembles for uncertainty estimation?
• RQ5 What impact do encoder architecture and density type (Mixture of Gaussians, radial flow, IAF) have on performance?

Key findings

• PostNet achieves competitive accuracy and confidence calibration without OOD data during training.
• PostNet significantly improves OOD detection compared to Dirichlet-based baselines, including Prior Networks.
• Flow-based density estimation in latent space yields strong OOD and calibration performance across datasets.
• Ablations show the Bayesian loss and density estimation are crucial for robust uncertainty estimation and OOD discrimination.
• Results on MNIST, CIFAR-10, and Sensorless Drive demonstrate good uncertainty estimates across architectures and data regimes.