# Graph Posterior Network: Bayesian Predictive Uncertainty for Node Classification

arXiv: https://arxiv.org/abs/2110.14012 · HTML: https://mlsanity.com/paper/2110.14012/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- Builds on 73 of its 116 references that are in the map.
- First posted on arXiv 26 Oct 2021.

## Builds on (most cited first, 30 of 73)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Evidential Deep Learning to Quantify Classification Uncertainty](https://mlsanity.com/paper/1806.01768/) (2018)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling](https://mlsanity.com/paper/1801.10247/) (2018)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Adversarial Examples Are Not Easily Detected: Bypassing Ten Detection Methods](https://mlsanity.com/paper/1705.07263/) (2017)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [Adversarial Attacks on Neural Networks for Graph Data](https://mlsanity.com/paper/1805.07984/) (2018)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning](https://mlsanity.com/paper/2002.06715/) (2020)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)

## Built on by (most cited first, 30 of 80)

- [Uncertainty Quantification over Graph with Conformalized Graph Neural Networks](https://mlsanity.com/paper/2305.14535/) (2023)
- [Energy-based Out-of-Distribution Detection for Graph Neural Networks](https://mlsanity.com/paper/2302.02914/) (2023)
- [A Survey of Graph Neural Networks in Real world: Imbalance, Noise, Privacy and OOD Challenges](https://mlsanity.com/paper/2403.04468/) (2024)
- [GOOD-D: On Unsupervised Graph Out-Of-Distribution Detection](https://mlsanity.com/paper/2211.04208/) (2022)
- [What Makes Graph Neural Networks Miscalibrated?](https://mlsanity.com/paper/2210.06391/) (2022)
- [Class-Imbalanced Graph Learning without Class Rebalancing](https://mlsanity.com/paper/2308.14181/) (2023)
- [Disentangling Epistemic and Aleatoric Uncertainty in Reinforcement Learning](https://mlsanity.com/paper/2206.01558/) (2022)
- [Uncertainty in Graph Neural Networks: A Survey](https://mlsanity.com/paper/2403.07185/) (2024)
- [Random-Set Neural Networks (RS-NN)](https://mlsanity.com/paper/2307.05772/) (2023)
- [AMIGO: Sparse Multi-Modal Graph Transformer with Shared-Context Processing for Representation Learning of Giga-pixel Images](https://mlsanity.com/paper/2303.00865/) (2023)
- [Graph Learning under Distribution Shifts: A Comprehensive Survey on Domain Adaptation, Out-of-distribution, and Continual Learning](https://mlsanity.com/paper/2402.16374/) (2024)
- [Calibrating Multimodal Learning](https://mlsanity.com/paper/2306.01265/) (2023)
- [GOODAT: Towards Test-time Graph Out-of-Distribution Detection](https://mlsanity.com/paper/2401.06176/) (2024)
- [JuryGCN: Quantifying Jackknife Uncertainty on Graph Convolutional Networks](https://mlsanity.com/paper/2210.05959/) (2022)
- [GOLD: Graph Out-of-Distribution Detection via Implicit Adversarial Latent Generation](https://mlsanity.com/paper/2502.05780/) (2025)
- [You Can Have Better Graph Neural Networks by Not Training Weights at All: Finding Untrained GNNs Tickets](https://mlsanity.com/paper/2211.15335/) (2022)
- [Uncertainty-Aware Graph Neural Networks: A Multi-Hop Evidence Fusion Approach](https://mlsanity.com/paper/2506.13083/) (2025)
- [Towards OOD Detection in Graph Classification from Uncertainty Estimation Perspective](https://mlsanity.com/paper/2206.10691/) (2022)
- [Graph Neural Networks in Particle Physics: Implementations, Innovations, and Challenges](https://mlsanity.com/paper/2203.12852/) (2022)
- [Robust Node Classification on Graphs: Jointly from Bayesian Label Transition and Topology-based Label Propagation](https://mlsanity.com/paper/2208.09779/) (2022)
- [Uncertainty for Active Learning on Graphs](https://mlsanity.com/paper/2405.01462/) (2024)
- [Uncertainty Estimation for Molecules: Desiderata and Methods](https://mlsanity.com/paper/2306.14916/) (2023)
- [Uncertainty Estimation on Sequential Labeling via Uncertainty Transmission](https://mlsanity.com/paper/2311.08726/) (2023)
- [Conformal Inductive Graph Neural Networks](https://mlsanity.com/paper/2407.09173/) (2024)
- [Decoupled Graph Energy-based Model for Node Out-of-Distribution Detection on Heterophilic Graphs](https://mlsanity.com/paper/2502.17912/) (2025)
- [Position: Epistemic uncertainty estimation methods are fundamentally incomplete](https://mlsanity.com/paper/2505.23506/) (2025)
- [Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations](https://mlsanity.com/paper/2408.16115/) (2024)
- [Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'](https://mlsanity.com/paper/2505.04950/) (2025)
- [Evaluating Robustness and Uncertainty of Graph Models Under Structural Distributional Shifts](https://mlsanity.com/paper/2302.13875/) (2023)
- [Training, Architecture, and Prior for Deterministic Uncertainty Methods](https://mlsanity.com/paper/2303.05796/) (2023)

## Abstract (arXiv)

The interdependence between nodes in graphs is key to improve class predictions on nodes and utilized in approaches like Label Propagation (LP) or in Graph Neural Networks (GNN). Nonetheless, uncertainty estimation for non-independent node-level predictions is under-explored. In this work, we explore uncertainty quantification for node classification in three ways: (1) We derive three axioms explicitly characterizing the expected predictive uncertainty behavior in homophilic attributed graphs. (2) We propose a new model Graph Posterior Network (GPN) which explicitly performs Bayesian posterior updates for predictions on interdependent nodes. GPN provably obeys the proposed axioms. (3) We extensively evaluate GPN and a strong set of baselines on semi-supervised node classification including detection of anomalous features, and detection of left-out classes. GPN outperforms existing approaches for uncertainty estimation in the experiments.