In this project we applied techniques from Random Geometric Graphs theory and Computational Topology to Graphical Models.
We worked within the framework proposed by Giudici and Green in 1999. In their paper, they proposed the following approach: let us represent a multivariate distribution as an entity composed of three factors. The first factor represents the distribution of the observations (think of samples from a multivariate normal for the sake of concreteness) given a collection of conditional independence statements (encoded by a graph) and a parameter that encodes the strength of the dependencies (for instance, the precision matrix of a multivariate normal). The second factor is the conditional distribution of the parameter encoding the strength of the dependencies given the graph; this term is known as the Hyper-Markov law. The third factor is the prior distribution on the graph space.
The main contribution of our work is to use random geometric graph models as priors for the space of graphs in graphical models. This can be achieved by identifying each marginal of the multivariate distribution of interest to a point in d-dimensional Euclidean space. By placing a convex set on top of each of these points or vertices, a graph or hypergraph is computed from the corresponding intersection pattern (this is known as a nerve). The Random Geometric Graphs theory gives us insight regarding the prior that is induced on graph space via this procedure, while the Computational Topology provides tools for computing these graphs and hypergraphs efficiently.