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Mathematics
Mathematical Logic

First-order Bayesian logic

Profile image of Kathryn LaskeyKathryn Laskey

2005

Abstract

Uncertainty is a fundamental and irreducible aspect of our knowledge about the world. Until recently, classical first-order logic has reigned as the de facto standard logical foundation for artificial intelligence. The lack of a built-in, semantically grounded capability for reasoning under uncertainty renders classical first-order logic inadequate for many important classes of problems. General-purpose languages are beginning to emerge for which the fundamental logical basis is probability. Increasingly expressive probabilistic languages demand a theoretical foundation that fully integrates classical first-order logic and probability. In first-order Bayesian logic (FOBL), probability distributions are defined over interpretations of classical first-order axiom systems. Predicates and functions of a classical first-order theory correspond to a random variables in the corresponding first-order Bayesian theory. This is a natural correspondence, given that random variables are formalized in mathematical statistics as measurable functions on a probability space. A formal system called Multi-Entity Bayesian Networks (MEBN) is presented for composing distributions on interpretations by instantiating and combining parameterized fragments of directed graphical models. A construction is given of a MEBN theory that assigns a non-zero probability to any satisfiable sentence in classical first-order logic. By conditioning this distribution on consistent sets of sentences, FOBL can represent a probability distribution over interpretations of any finitely axiomatizable first-order theory, as well as over interpretations of infinite axiom sets when a limiting distribution exists. FOBL is inherently open, having the ability to incorporate new axioms into existing theories, and to modify probabilities in the light of evidence. Bayesian inference provides both a proof theory for combining prior knowledge with observations, and a learning theory for refining a representation as evidence accrues. The results of this paper provide a logical foundation for the rapidly evolving literature on first-order Bayesian knowledge representation, and point the way toward Bayesian languages suitable for generalpurpose knowledge representation and computing. Because FOBL contains classical first-order logic as a deterministic subset, it is a natural candidate as a universal representation for integrating domain ontologies expressed in languages based on classical first-order logic or subsets thereof.

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