# Bayes-Ball: The Rational Pastime (for Determining Irrelevance and Requisite Information in Belief Networks and Influence Diagrams)

arXiv: https://arxiv.org/abs/1301.7412 · HTML: https://mlsanity.com/paper/1301.7412/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2013.

## Built on by (most cited first, 18 of 18)

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- [Adjustment Identification Distance: A gadjid for Causal Structure Learning](https://mlsanity.com/paper/2402.08616/) (2024)
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- [Mean-Field Sampling for Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2412.00661/) (2024)
- [A Complete Criterion for Value of Information in Soluble Influence Diagrams](https://mlsanity.com/paper/2202.11629/) (2022)
- [Linear-Time Algorithms for Front-Door Adjustment in Causal Graphs](https://mlsanity.com/paper/2211.16468/) (2022)
- [On Imperfect Recall in Multi-Agent Influence Diagrams](https://mlsanity.com/paper/2307.05059/) (2023)
- [Differentiable Constraint-Based Causal Discovery](https://mlsanity.com/paper/2510.22031/) (2025)
- [Estimating Causal Effects in Partially Directed Parametric Causal Factor Graphs](https://mlsanity.com/paper/2411.07006/) (2024)
- [Containing a spread through sequential learning: to exploit or to explore?](https://mlsanity.com/paper/2303.00141/) (2023)
- [Testing Causal Models with Hidden Variables in Polynomial Delay via Conditional Independencies](https://mlsanity.com/paper/2409.14593/) (2024)
- [A Graphical Approach to State Variable Selection in Off-policy Learning](https://mlsanity.com/paper/2501.00854/) (2025)
- [Towards Causal Representation Learning with Observable Sources as Auxiliaries](https://mlsanity.com/paper/2509.19058/) (2025)
- [First-Order Context-Specific Likelihood Weighting in Hybrid Probabilistic Logic Programs](https://mlsanity.com/paper/2201.11165/) (2022)
- [Stabilizing Subject Transfer in EEG Classification with Divergence Estimation](https://mlsanity.com/paper/2310.08762/) (2023)
- [Mechanisms for Data Sharing in Collaborative Causal Inference (Extended Version)](https://mlsanity.com/paper/2407.11032/) (2024)
- [Linear-Time Primitives for Algorithm Development in Graphical Causal Inference](https://mlsanity.com/paper/2506.15758/) (2025)
- [Lifted Causal Inference](https://mlsanity.com/paper/2606.28024/) (2026)

## Abstract (arXiv)

One of the benefits of belief networks and influence diagrams is that so much knowledge is captured in the graphical structure. In particular, statements of conditional irrelevance (or independence) can be verified in time linear in the size of the graph. To resolve a particular inference query or decision problem, only some of the possible states and probability distributions must be specified, the "requisite information." This paper presents a new, simple, and efficient "Bayes-ball" algorithm which is well-suited to both new students of belief networks and state of the art implementations. The Bayes-ball algorithm determines irrelevant sets and requisite information more efficiently than existing methods, and is linear in the size of the graph for belief networks and influence diagrams.