From Probabilistic Horn Logic to Chain Logic

Probabilistic logics have attracted a great deal of attention during the past few years. Where logical languages have, already from the inception of the field of artificial intelligence, taken a central position in research on knowledge representation and automated reasoning, probabilistic graphical models with their associated probabilistic basis have taken up in recent years a similar position when it comes to reasoning with uncertainty. There are now several different proposals in literature to merge logic and probabilistic graphical models. Probabilistic Horn logic combines Horn logic with probability theory, which yields a probabilistic logic that allows reasoning with classes of Bayesian networks. Bayesian logic is similar in expressive power to probabilistic Horn logic; the main difference is that it is primarily meant as a language for generating Bayesian networks. Finally, Markov logic networks have recently been proposed as a language for generating Markov networks using a model-theoretic interpretation of a logical specification. However, whereas Bayesian networks have an attractive semantics, they suffer from the fact that different Bayesian networks may represent exactly the same independence relation. Markov networks, on the other hand, lack in expressiveness when representing independence information. The formalism of chain graphs is increasingly seen as a natural probabilistic graphical formalism as it generalises both Bayesian networks and Markov networks, and has an attractive semantics in the sense that any Bayesian network has a unique graphical representation as a chain graph. In this paper, a new probabilistic logic, called chain logic, is developed along the lines of probabilistic Horn logic. This new probabilistic logic allows representing subtle independence information that cannot be represented by all previously developed probabilistic logics.