Bayesian networks (BNs) are graphical first-order probabilistic models that allow for a compact representation of large probability distributions, and efficient inference, both exact and approximate. We introduce a higher-order programming language, in the idealized form of a lambda-calculus, which we prove sound and complete w.r.t. BNs: each BN can be encoded as a term, and conversely each (possibly higher-order and recursive) program of ground type compiles into a BN. Moreover, we provide a compositional and cost-aware semantics which is based on factors, the standard mathematical tool used in Bayesian inference. Our results rely on advanced techniques rooted into linear logic, intersection types, rewriting theory, and Girard’s geometry of interaction, which are here combined in a novel way.
