On the Foundations of Cycles in Bayesian Networks
This work addresses a foundational issue in probabilistic graphical models for researchers and practitioners dealing with feedback loops, but it is incremental as it builds on existing cycle-handling methods.
The paper tackles the problem of directed cycles in Bayesian networks, which naturally arise from cross-dependencies like feedback loops, by proposing foundational semantics for cyclic BNs that conservatively extend the cycle-free setting, including constraint-based and limit semantics shown to be computable via a Markov chain construction.
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.