On the Foundations of Cycles in Bayesian Networks

TL;DR

This paper explores the foundational semantics of cyclic Bayesian networks, extending traditional acyclic models to include feedback loops through constraint-based and limit semantics, ensuring consistency and computability.

Contribution

It introduces a formal semantics framework for cyclic Bayesian networks, including constraint-based and limit semantics, to handle feedback loops while maintaining probabilistic consistency.

Findings

Constraint-based semantics specify consistency requirements for joint distributions.

Limit semantics formalize infinite unfolding of cyclic networks.

Markov chain construction enables computability of limit semantics.

Abstract

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.