Bayesian Network Automata for Modelling Unbounded Structures

This paper proposes a framework which unifies graphical model theory and formal language theory through automata theory. Specifically, we propose Bayesian Network Automata (BNAs) as a formal framework for specifying graphical models of arbitrarily large structures, or equivalently, specifying probabilistic grammars in terms of graphical models. BNAs use a formal automaton to specify how to construct an arbitrarily large Bayesian Network by connecting multiple copies of a bounded Bayesian Network. Using a combination of results from graphical models and formal language theory, we show that, for a large class of automata, the complexity of inference with a BNA is bounded by the complexity of inference in the bounded Bayesian Network times the complexity of inference for the equivalent stochastic automaton. This illustrates that BNAs provide a useful framework for developing and analysing models and algorithms for structure prediction.