Greedy learning of Markov network structure
Markov Random Fields (MRFs), a.k.a. Graphical Models, serve as popular models for networks in the social and biological sciences, as well as communications and signal processing. A central problem is one of structure learning or model selection: given samples from the MRF, determine the graph structure of the underlying distribution. When the MRF is not Gaussian (e.g. the Ising model) and contains cycles, structure learning is known to be NP hard even with infinite samples. Existing approaches typically focus either on specific parametric classes of models, or on the sub-class of graphs with bounded degree; the complexity of many of these methods grows quickly in the degree bound. We develop a simple new ‘greedy’ algorithm for learning the structure of graphical models of discrete random variables. It learns the Markov neighborhood of a node by sequentially adding to it the node that produces the highest reduction in conditional entropy. We provide a general sufficient condition for exact structure recovery (under conditions on the degree/girth/correlation decay), and study its sample and computational complexity. We then consider its implications for the Ising model, for which we establish a self-contained condition for exact structure recovery.