  We introduce Probabilistic Dependency Graphs (PDGs), a new class of directed
graphical models. PDGs can capture inconsistent beliefs in a natural way and
are more modular than Bayesian Networks (BNs), in that they make it easier to
incorporate new information and restructure the representation. We show by
example how PDGs are an especially natural modeling tool. We provide three
semantics for PDGs, each of which can be derived from a scoring function (on
joint distributions over the variables in the network) that can be viewed as
representing a distribution's incompatibility with the PDG. For the PDG
corresponding to a BN, this function is uniquely minimized by the distribution
the BN represents, showing that PDG semantics extend BN semantics. We show
further that factor graphs and their exponential families can also be
faithfully represented as PDGs, while there are significant barriers to
modeling a PDG with a factor graph.
