The Descriptive Complexity of Bayesian Network Specifications

Fábio Gagliardi Cozman, Denis Deratani Mauá

Abstract: We adapt the theory of descriptive complexity to Bayesian networks, by investigating how expressive can be specifications based on predicates and quantifiers. We show that Bayesian network specifications that employ first-order quantification capture the complexity class \(\mathsf {PP}\); that is, any phenomenon that can be simulated with a polynomial time probabilistic Turing machine can be also modeled by such a network. We also show that, by allowing quantification over predicates, the resulting Bayesian network specifications capture the complexity class \(\mathsf {PP}^\mathsf {NP}\), a result that does not seem to have equivalent in the literature.