Probabilistic Theorem Proving

Abstract

Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logical structure into account. We propose the first method that has the full power of both graphical model inference and first-order theorem proving (in finite domains with Herbrand interpretations). We first define probabilistic theorem proving, their generalization, as the problem of computing the probability of a logical formula given the probabilities or weights of a set of formulas. We then show how this can be reduced to the problem of lifted weighted model counting, and develop an efficient algorithm for the latter. We prove the correctness of this algorithm, investigate its properties, and show how it generalizes previous approaches. Experiments show that it greatly outperforms lifted variable elimination when logical structure is present. Finally, we propose an algorithm for approximate probabilistic theorem proving, and show that it can greatly outperform lifted belief propagation.

References in corpus (4)

• Counting Belief Propagation
• Bisimulation-based Approximate Lifted Inference
• Constraint Processing in Lifted Probabilistic Inference
• Formula-Based Probabilistic Inference

Cited by in corpus (9)

• Markov Chains on Orbits of Permutation Groups
• Automorphism Groups of Graphical Models and Lifted Variational Inference
• Exploiting Uniform Assignments in First-Order MPE
• Markov Logic Networks for Natural Language Question Answering
• Probabilistic Databases with MarkoViews
• Lifted Variable Elimination: A Novel Operator and Completeness Results
• Approximate Lifted Inference with Probabilistic Databases
• Symmetric Weighted First-Order Model Counting
• Lifted Region-Based Belief Propagation