1 Markov Logic: A Unifying Framework for Statistical Relational Learning

Interest in statistical relational learning (SRL) has grown rapidly in recent years. Several key SRL tasks have been identified, and a large number of approaches have been proposed. Increasingly, a unifying framework is needed to facilitate transfer of knowledge across tasks and approaches, to compare approaches, and to help bring structure to the field. We propose Markov logic as such a framework. Syntactically, Markov logic is indistinguishable from first-order logic, except that each formula has a weight attached. Semantically, a set of Markov logic formulas represents a probability distribution over possible worlds, in the form of a log-linear model with one feature per grounding of a formula in the set, with the corresponding weight. We show how approaches like probabilistic relational models, knowledgebased model construction and stochastic logic programs can be mapped into Markov logic. We also show how tasks like collective classification, link prediction, linkbased clustering, social network modeling, and object identification can be concisely formulated in Markov logic. Finally, we develop learning and inference algorithms for Markov logic, and report experimental results on a link prediction task. 1.1 The Need for a Unifying Framework Many (if not most) real-world application domains are characterized by the presence of both uncertainty and complex relational structure. Statistical learning focuses 2 Markov Logic: A Unifying Framework for Statistical Relational Learning on the former, and relational learning on the latter. Statistical relational learning (SRL) seeks to combine the power of both. Research in SRL has expanded rapidly in recent years, both because of the need for it in applications, and because statistical and relational learning have individually matured to the point where combining them is a feasible research enterprise. A number of key SRL tasks have been identified, including collective classification, link prediction, link-based clustering, social network modeling, object identification, and others. A large and growing number of SRL approaches have been proposed, including knowledge-based model construction [Wellman et al., 1992, Ngo and Haddawy, 1997, Kersting and De Raedt, 2001], stochastic logic programs [Muggleton, 1996, Cussens, 1999], PRISM [Sato and Kameya, 1997], MACCENT [Dehaspe, 1997], probabilistic relational models [Friedman et al., 1999], relational Markov models [Anderson et al., 2002], relational Markov networks [Taskar et al., 2002], relational dependency networks [Neville and Jensen, 2003], structural logistic regression [Popescul and Ungar, 2003], relational generation functions [Cumby and Roth, 2003], CLP(BN ) [Santos Costa et al., 2003], and others. While the variety of problems and approaches in the field is valuable, it makes it difficult for researchers, students and practitioners to identify, learn and apply the essentials. In particular, for the most part, the relationships between different approaches and their relative strengths and weaknesses remain poorly understood, and innovations in one task or application do not easily transfer to others, slowing down progress. There is thus an increasingly pressing need for a unifying framework, a common language for describing and relating the different tasks and approaches. To be most useful, such a framework should satisfy the following desiderata: 1. The framework must incorporate both first-order logic and probabilistic graphical models. Otherwise some current or future SRL approaches will fall outside its scope. 2. SRL problems should be representable clearly and simply in the framework. 3. The framework must facilitate the use of domain knowledge in SRL. Because the search space for SRL algorithms is very large even by AI standards, domain knowledge is critical to success. Conversely, the ability to incorporate rich domain knowledge is one of the most attractive features of SRL. 4. The framework should facilitate the extension to SRL of techniques from statistical learning, inductive logic programming, probabilistic inference and logical inference. This will speed progress in SRL by taking advantage of the large extant literature in these areas. In this chapter we propose Markov logic as a framework that we believe meets all of these desiderata. We begin by briefly reviewing the necessary background in Markov networks (Section 1.2) and first-order logic (Section 1.3). We then introduce Markov logic (Section 1.4) and describe how several SRL approaches and tasks can be formulated in this framework (Sections 1.5 and 1.6). Next, we show how techniques from logic, probabilistic inference, statistics and inductive logic programming can be used to obtain practical inference and learning algorithms for Markov logic 1.2 Markov Networks 3 (Sections 1.7 and 1.8). Finally, we illustrate the application of these algorithms in a real-world link prediction task (Section 1.9) and conclude (Section 1.10).