In this paper we propose a class of prior distributions on decompos-
able graphs, allowing for improved modeling
exibility. While existing methods
solely penalize the number of edges, the proposed work empowers practitioners to
control clustering, level of separation, and other features of the graph. Emphasis
is placed on a particular prior distribution which derives its motivation from the
class of product partition models; the properties of this prior relative to existing
priors are examined through theory and simulation. We then demonstrate the
use of graphical models in the eld of agriculture, showing how the proposed prior
distribution alleviates the in
exibility of previous approaches in properly modeling
the interactions between the yield of di erent crop varieties. Lastly, we explore
American voting data, comparing the voting patterns amongst the states over the
last century.