We consider three desiderata for a language combining logic and probability: logical expressivity, random-world semantics, and the existence of a useful syntactic condition for probabilistic independence. Achieving these three desiderata simultaneously is nontrivial. Expressivity can be achieved by using a formalism similar to a programming language, but standard approaches to combining programming languages with probabilities sacrifice random-world semantics. Naive approaches to restoring random-world semantics undermine syntactic independence criteria. Our main result is a syntactic independence criterion that holds for a broad class of highly expressive logics under random-world semantics. We explore various examples including Bayesian networks, probabilistic context-free grammars, and an example from Mendelian genetics. Our independence criterion supports a case-factor inference technique that reproduces both variable elimination for BNs and the inside algorithm for PCFGs.
