4. Relational Languages: Inferential, query, and domain complexity
In this section we extend our specification framework so as to deal with relational languages. Such languages have been used in a variety of applications with repetitive entities and relationships [46, 109], as we have alluded to in Section 1. They have roots in general probabilistic logics that mix deterministic knowledge and uncertain reasoning in very flexible, but sometimes too complicated, ways [56]. The more focused interest in extending Bayesian networks with relational constructs has produced an array of practical languages, as we summarize later in Section 7: plates, PRMs, probabilistic logic programs. It is not easy to extract a “common denominator” from these languages. However, with some reflection we see that they all parameterize random variables using relations; they all allow for logical definitions to be mixed with probabilistic assessments; they all resort to the semantics of first-order logic to interpret relations using sets (domains). We capture these common features in Section 4.1; to do so, we use “parvariables” and related techniques introduced by Poole [103], using as much syntax and semantics as possible from first-order logic and in particular from lifted inference techniques [90] (and borrowing some terminology from description logics and logic programming as appropriate). Throughout we resort to the same idea advocated in Section 3: that is, that a model can be specified by a set of probability assessments and a set of logical definitions that belong to a selected logical language. This is exactly the situation in acyclic probabilistic logic programs and many other existing languages, as shown later in Section 7.
