Negative probabilities in probabilistic logic programs

Abstract

We consider probabilistic logic programs (PLPs) for non-extreme distributions. We show that in the relational case with fixed populations, PLPs cannot represent many non-extreme distributions, even using negations. We introduce negative rule probabilities in PLPs, and show they make the language strictly more expressive. In addition, they render negations unnecessary: negations in PLPs can be translated to rules with negative probabilities, thus avoiding the problem of logical inconsistency. Furthermore, this translation keeps the PLP size compact (assuming the number of negations per rule is small). This translation algorithm also allows algorithms for exact inference that do not support negations to be applicable to PLPs with negations.The noise probabilities for non-exclusive rules are difficult to interpret and unintuitive to manipulate. To alleviate this we define “probability-strengths”, an alternative representation for probabilistic values, which results in an intuitive additive algebra for combining rules.For acyclic propositional PLPs we prove what constraints on the strengths allow for proper distributions on the non-noise variables and allow for all non-extreme distributions to be represented. We show how arbitrary CPDs can be converted into this form in a canonical way.