Bounds on the number of inference functions of a graphical model

Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of inference functions of any graphical model. This bound is polynomial on the size of the model, for a fixed number of parameters. We also show that our bound is tight up to a constant factor, by constructing a family of hidden Markov models whose number of inference functions agrees asymptotically with the upper bound. This paper elaborates and expands on results of the first author from Elizalde (2005).