[Paper Review] Bayesian Logic Programs

Kristian Kersting, Luc De Raedt|Lirias|Nov 23, 2001

Bayesian Modeling and Causal InferenceComputer Science42 references167 citations

TL;DR

This paper introduces Bayesian Logic Programs (BLPs), a framework that unifies Bayesian networks and definite clause logic by establishing a one-to-one mapping between ground atoms and random variables. By combining the qualitative structure of logic programs with the quantitative reasoning of Bayesian networks, BLPs enable scalable, modular modeling of relational and probabilistic domains, including dynamic systems and continuous variables, while ensuring computable inference even over infinite Herbrand bases.

ABSTRACT

Bayesian networks provide an elegant formalism for representing and reasoning about uncertainty using probability theory. Theyare a probabilistic extension of propositional logic and, hence, inherit some of the limitations of propositional logic, such as the difficulties to represent objects and relations. We introduce a generalization of Bayesian networks, called Bayesian logic programs, to overcome these limitations. In order to represent objects and relations it combines Bayesian networks with definite clause logic by establishing a one-to-one mapping between ground atoms and random variables. We show that Bayesian logic programs combine the advantages of both definite clause logic and Bayesian networks. This includes the separation of quantitative and qualitative aspects of the model. Furthermore, Bayesian logic programs generalize both Bayesian networks as well as logic programs. So, many ideas developed

Motivation & Objective

• To address the limitations of propositional Bayesian networks in modeling relational and dynamic domains with variable numbers of objects.
• To overcome the rigidity of propositional representations by integrating first-order logic with probabilistic graphical models.
• To enable the representation of general probabilistic regularities across instances (e.g., families, computer networks) rather than per-instance models.
• To provide a formalism that separates qualitative (structural) and quantitative (probabilistic) components, preserving modularity and reusability.
• To support inference over infinite or large domains through a sound, computable query-answering procedure based on knowledge-based model construction.

Proposed method

• Establish a one-to-one correspondence between ground atoms in definite clause logic and random variables in a Bayesian network.
• Represent the qualitative structure of the model using logic programs (definite clauses), where each clause defines a conditional probability distribution.
• Define the joint probability distribution over all ground atoms via the least Herbrand model, ensuring a unique, well-defined Bayesian network structure.
• Use a two-phase query-answering procedure: (1) construct a support network from the logical derivation (AND/OR graph), and (2) apply Bayesian inference on the resulting network.
• Leverage SLD resolution and AND/OR graphs to guide inference, enabling efficient computation even when the Herbrand base is infinite.
• Implement a Prolog-based meta-interpreter to perform knowledge-based model construction and support practical execution.

Experimental results

Research questions

• RQ1 Can a formalism be designed that combines the relational expressiveness of logic programs with the uncertainty reasoning of Bayesian networks?
• RQ2 How can a one-to-one mapping between ground atoms and random variables be used to define a coherent, finite or infinite Bayesian network?
• RQ3 What is the computational complexity of querying a Bayesian logic program, and can inference remain tractable even when the Herbrand base is infinite?
• RQ4 How can dynamic and continuous-variable models be represented within a first-order probabilistic framework?
• RQ5 What is the relationship between Bayesian logic programs and existing first-order probabilistic models such as dynamic Bayesian networks and stochastic logic programs?

Key findings

• Bayesian Logic Programs generalize both Bayesian networks and definite clause logic, allowing the modeling of any Bayesian network and any Prolog program within a single framework.
• The framework ensures a strict separation between qualitative (logical) and quantitative (probabilistic) components, enhancing modularity and interpretability.
• Even when the least Herbrand model is infinite, every well-defined probabilistic query remains computable due to the finite support of queries.
• The query-answering procedure is based on a two-phase strategy: support network construction via SLD resolution and inference on the resulting Bayesian network.
• The approach naturally supports dynamic models, including dynamic Bayesian networks and hidden Markov models, by encoding temporal dependencies in the logic program.
• A simple Prolog meta-interpreter is implemented that enables practical execution and knowledge-based model construction for BLPs.