# On the Foundations of Cycles in Bayesian Networks

arXiv: https://arxiv.org/abs/2301.08608 · HTML: https://mlsanity.com/paper/2301.08608/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 33 references that are in the map.
- First posted on arXiv 20 Jan 2023.

## Builds on (most cited first, 6 of 6)

- [Markov Properties for Graphical Models with Cycles and Latent Variables](https://mlsanity.com/paper/1710.08775/) (2017)
- [Directed Cyclic Graphical Representations of Feedback Models](https://mlsanity.com/paper/1302.4982/) (2013)
- [Identifying Independencies in Causal Graphs with Feedback](https://mlsanity.com/paper/1302.3595/) (2013)
- [Bayesian Attack Model for Dynamic Risk Assessment](https://mlsanity.com/paper/1606.09042/) (2016)
- [Structure learning of Bayesian networks involving cyclic structures](https://mlsanity.com/paper/1906.04992/) (2019)
- [Cyclic Bayesian Attack Graphs: A Systematic Computational Approach](https://mlsanity.com/paper/2005.06350/) (2020)

## Built on by (most cited first, 2 of 2)

- [Causal Concept Graph Models: Beyond Causal Opacity in Deep Learning](https://mlsanity.com/paper/2405.16507/) (2024)
- [Markov Conditions and Factorization in Logical Credal Networks](https://mlsanity.com/paper/2302.14146/) (2023)

## Abstract (arXiv)

Bayesian networks (BNs) are a probabilistic graphical model widely used for representing expert knowledge and reasoning under uncertainty. Traditionally, they are based on directed acyclic graphs that capture dependencies between random variables. However, directed cycles can naturally arise when cross-dependencies between random variables exist, e.g., for modeling feedback loops. Existing methods to deal with such cross-dependencies usually rely on reductions to BNs without cycles. These approaches are fragile to generalize, since their justifications are intermingled with additional knowledge about the application context. In this paper, we present a foundational study regarding semantics for cyclic BNs that are generic and conservatively extend the cycle-free setting. First, we propose constraint-based semantics that specify requirements for full joint distributions over a BN to be consistent with the local conditional probabilities and independencies. Second, two kinds of limit semantics that formalize infinite unfolding approaches are introduced and shown to be computable by a Markov chain construction.