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, thus improving the exponential upper bound given by Pachter and Sturmfels [14] . 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. Finally, we apply this bound to a model for sequence alignment that is used in computational biology.
A study which is complementary to the upper bound on the size of envelopes derived in Section 4 is provided in REF which shows that the number of inference functions of any graphical model as, for instance, Bayesian networks and Markov random fields is polynomial in the size of the model if the number of parameters is fixed.
Bounds on the number of inference functions of a graphical model
Formal Power Series and Algebraic Combinatorics (FPSAC 18
Mathematics Biology Computer Science