| Graphical models is the association of a graph with a multivariate distribution, where each node in the graph is associated with one of the random variables in the distribution. An edge exists between two nodes if and only if the random variables associated with the two nodes are conditionally independent given the rest of the nodes. Graphical model learning is the learning of the topology of this associated graph based on samples of the distribution. This may include learning the distribution's model parameters as well.;In this dissertation, we will explore graphical learning for the multivariate Gaussian distribution and the Ising distribution. One of the more common ways of making graphical model learning easier to assume that the graph is degree bounded, making the graph sparse. In our discussion, we will instead look at the idea of separability instead. We do not restrict the number of neighbors of the nodes, rather, we will limit the number of vertex disjoint paths between pairs of nodes. We believe this is more natural way of describing the relationship between pairs of nodes in a graphical model.;A weakly K-separable graph is a graph where all non-neighbor pairs of nodes are restricted in this way: there are at most K vertex disjoint paths between any two non-neighboring nodes. In a Markov random field, which includes the Gaussian and Ising models we are examining, the separability of two nodes implies conditional independence via the Global Markov Property. The converse, however, does not hold. However, we are able to test for when it does. Using this test, we provide a novel graph learning algorithm for weakly K-separable Gaussian graphical models.;We will define other notions of separability as well, namely, strongly K-separability, (K,L)-separability and neighbor L-separability. We introduced graph learning algorithms for these classes of graphical models as well. Finally, we will take a look at the converse of the Global Markov Property for the Ising model, and study when the converse does and does not hold. |
