  We present a framework for incorporating prior information into nonparametric
estimation of graphical models. To avoid distributional assumptions, we
restrict the graph to be a forest and build on the work of forest density
estimation (FDE). We reformulate the FDE approach from a Bayesian perspective,
and introduce prior distributions on the graphs. As two concrete examples, we
apply this framework to estimating scale-free graphs and learning multiple
graphs with similar structures. The resulting algorithms are equivalent to
finding a maximum spanning tree of a weighted graph with a penalty term on the
connectivity pattern of the graph. We solve the optimization problem via a
minorize-maximization procedure with Kruskal's algorithm. Simulations show that
the proposed methods outperform competing parametric methods, and are robust to
the true data distribution. They also lead to improvement in predictive power
and interpretability in two real data sets.
