  An undirected graphical model is a joint probability distribution defined on
an undirected graph G*, where the vertices in the graph index a collection of
random variables and the edges encode conditional independence relationships
among random variables. The undirected graphical model selection (UGMS) problem
is to estimate the graph G* given observations drawn from the undirected
graphical model. This paper proposes a framework for decomposing the UGMS
problem into multiple subproblems over clusters and subsets of the separators
in a junction tree. The junction tree is constructed using a graph that
contains a superset of the edges in G*. We highlight three main properties of
using junction trees for UGMS. First, different regularization parameters or
different UGMS algorithms can be used to learn different parts of the graph.
This is possible since the subproblems we identify can be solved independently
of each other. Second, under certain conditions, a junction tree based UGMS
algorithm can produce consistent results with fewer observations than the usual
requirements of existing algorithms. Third, both our theoretical and
experimental results show that the junction tree framework does a significantly
better job at finding the weakest edges in a graph than existing methods. This
property is a consequence of both the first and second properties. Finally, we
note that our framework is independent of the choice of the UGMS algorithm and
can be used as a wrapper around standard UGMS algorithms for more accurate
graph estimation.
