# Probabilistic Modelling of Operational Design Domains, A New Approach for Testing AI Systems

arXiv: https://arxiv.org/abs/2609.24397 · HTML: https://mlsanity.com/paper/2609.24397/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 69 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Survey Of Methods For Explaining Black Box Models](https://mlsanity.com/paper/1802.01933/) (2018)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [DeepXplore: Automated Whitebox Testing of Deep Learning Systems](https://mlsanity.com/paper/1705.06640/) (2017)
- [Machine Learning Testing: Survey, Landscapes and Horizons](https://mlsanity.com/paper/1906.10742/) (2019)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [DeepGauge: Multi-Granularity Testing Criteria for Deep Learning Systems](https://mlsanity.com/paper/1803.07519/) (2018)
- [Introduction to Neural Network Verification](https://mlsanity.com/paper/2109.10317/) (2021)
- [Feature-Guided Black-Box Safety Testing of Deep Neural Networks](https://mlsanity.com/paper/1710.07859/) (2017)
- [Scenic: A Language for Scenario Specification and Data Generation](https://mlsanity.com/paper/2010.06580/) (2020)
- [Grid Saliency for Context Explanations of Semantic Segmentation](https://mlsanity.com/paper/1907.13054/) (2019)
- [Corner Cases for Visual Perception in Automated Driving: Some Guidance on Detection Approaches](https://mlsanity.com/paper/2102.05897/) (2021)
- [6-Layer Model for a Structured Description and Categorization of Urban Traffic and Environment](https://mlsanity.com/paper/2012.06319/) (2020)
- [One Ontology to Rule Them All: Corner Case Scenarios for Autonomous Driving](https://mlsanity.com/paper/2209.00342/) (2022)
- [Towards Dependability Metrics for Neural Networks](https://mlsanity.com/paper/1806.02338/) (2018)
- [A Comprehensive Survey on Imbalanced Data Learning](https://mlsanity.com/paper/2502.08960/) (2025)
- [Certified Control: An Architecture for Verifiable Safety of Autonomous Vehicles](https://mlsanity.com/paper/2104.06178/) (2021)
- [A Comprehensive Review on Ontologies for Scenario-based Testing in the Context of Autonomous Driving](https://mlsanity.com/paper/2304.10837/) (2023)
- [A Stochastic Approach to Classification Error Estimates in Convolutional Neural Networks](https://mlsanity.com/paper/2401.06156/) (2023)
- [Outline of an Independent Systematic Blackbox Test for ML-based Systems](https://mlsanity.com/paper/2401.17062/) (2024)

## Abstract (arXiv)

The conventional testing process quickly fails when applied to ML-based systems such as obstacle detection in vehicles: if an obstacle is not detected in a test, classical bug fixing is impossible and an AI system will always retain shortcomings. Test results can therefore only be interpreted statistically, which in turn requires test sets that are not only complete with respect to the operational design domain (ODD) of the system, but also representative of it. To this end, we introduce probabilistically extended ontologies (PEONs): ontologies describing the ODD, augmented with a probability distribution over the partitioning they induce. Instead of unmaintainable conditional probability tables, only marginal distributions and functionally described dependencies need to be specified; algorithms based on couplings and optimal transport complete this specification to a Bayesian network. From a PEON we derive the sampling of representative test cases, rigorous end-of-test criteria for given quality targets and significance levels, and methods for re-evaluating existing test results and for assessing the balance of training data. We demonstrate the practical modelling of a complex ODD using the example of automatic train operation.