Probabilistic Theorem Proving
This addresses a fundamental bottleneck in AI for combining logic and probability, though it appears incremental as it builds on existing lifted inference approaches.
The paper tackles the problem of unifying logical and probabilistic inference by proposing the first method with full power of both graphical model inference and first-order theorem proving, showing it greatly outperforms existing methods like lifted variable elimination and lifted belief propagation when logical structure is present.
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.