Lebesgue Logic for Probabilistic Reasoning and Some Applications to Perception

Abstract

Reasoning with probabilities is essential to many sciences, such as decision theory, expert systems, neural networks, pattern recognition, and perception in general. In this paper we explore a new logic of probabilities, the Lebesgue logic, in which are defined the logical relations ENTAILS, AND, OR, and NOT on collections of probability measures. In particular, given any two probability measures μ and v, the Lebesgue logic answers questions such as the following: Does μ entail v? What is the conjunction of μ and v, i.e., what is μ AND v? What is the disjunction of μ and v, i.e., what is μ OR v? What is the negation of μ? Several properties of the Lebesgue logic emerge. Among them are (1) the Lebesgue logic is not boolean, in general, but is "locally boolean," (2) the AND of the Lebesgue logic is a generalization of Bayes' rule; (3) one can define probability measures on the Lebesgue logic itself, thereby permitting the representation of probabilistic knowledge without requiring any commitment to a particular probability measure; and (4) many probabilistic inferences can be described as morphisms of the Lebesgue logic, i.e., as maps from one collection of probability measures to another, respecting the Lebesgue logics on both. We close by discussing a concrete problem to which the Lebesgue logic may find application; the problem of sensor fusion in vision and other perceptual modalities.