### Logical and algorithmic properties of stable conditional independence

Mathias Niepert, Dirk Van Gucht, Marc Gyssens
2010 International Journal of Approximate Reasoning
The logical and algorithmic properties of stable conditional independence (CI) as an alternative structural representation of conditional independence information are investigated. We utilize recent results concerning a complete axiomatization of stable conditional independence relative to discrete probability measures to derive perfect model properties of stable conditional independence structures. We show that stable CI can be interpreted as a generalization of Markov networks and establish a
more » ... connection between sets of stable CI statements and propositional formulas in conjunctive normal form. Consequently, we derive that the implication problem for stable CI is coNP-complete. Finally, we show that Boolean satisfiability (SAT) solvers can be employed to efficiently decide the implication problem and to compute concise, non-redundant representations of stable CI, even for instances involving hundreds of random variables. Ó 2010 Elsevier Inc. All rights reserved. Introduction Conditional independence is an important concept in many calculi for dealing with knowledge and uncertainty in artificial intelligence. The notion plays a fundamental role for learning and reasoning in intelligent systems. A conditional independence (CI) statement speaks to the independence of two sets of random variables relative to a third: given three mutually disjoint sets A; B, and C of random variables, A and B are conditionally independent relative to C if any instantiation of the variables in C renders the variables in A and B independent. In other words, if we have knowledge about the state of C, then knowledge about the state of A does not provide additional evidence for the state of B and vice versa. We use the notation IðA; BjCÞ to specify this independence condition. When novel information becomes available in a probabilistic system, the set of associated, relevant CI statements changes dynamically. However, some of the CI statements will continue to hold, i.e., they remain stable under change in the system. Technically, the notion of stability of a CI statement IðA; BjCÞ, in the context of a set of random variables S and a set of CI statements C, is defined by requiring that, for every superset C 0 C which is disjoint from A and B, the CI statement IðA; BjC 0 Þ also holds. In other words, the independence of A and B relative to C is unaffected by adding random variables to C. Clearly, this property does not hold in general. Adding variables to the set C may affect the (in-)dependence of A and B. A special case for which the stability of the CI statement IðA; BjCÞ is guaranteed is the situation where BjCÞ is said to be saturated.) Among the most frequently used models for representing conditional independence information are graphs, wherein the nodes correspond to random variables and the edges encode the (in-)dependence information among the variables. The most 0888-613X/\$ -see front matter Ó