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On the Foundations of Cycles in Bayesian Networks

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Principles of Systems Design

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13660))


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.

This work was partially supported by the DFG in projects TRR 248 (CPEC, see, project ID 389792660) and EXC 2050/1 (CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), and the Key-Area Research and Development Program Grant 2018B010107004 of Guangdong Province.

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  1. 1.

    We use Boolean random variables for simplicity of representation, an extension of the proposed semantics over random variables with arbitrary finite state spaces is certainly possible.

  2. 2.

    A path is simple if no node occurs twice in the path. “Undirected” in this context means that edges in either direction can occur along the path.

  3. 3.

    Recall that we may view distributions as vectors which allows us to equate distributions over different but isomorphic domains.


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Correspondence to Nikolai Käfer .

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Baier, C., Dubslaff, C., Hermanns, H., Käfer, N. (2022). On the Foundations of Cycles in Bayesian Networks. In: Raskin, JF., Chatterjee, K., Doyen, L., Majumdar, R. (eds) Principles of Systems Design. Lecture Notes in Computer Science, vol 13660. Springer, Cham.

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