  Probabilistic graphical models are widely used to model complex systems under
uncertainty. Traditionally, Gaussian directed graphical models are applied for
analysis of large networks with continuous variables as they can provide
conditional and marginal distributions in closed form simplifying the
inferential task. The Gaussianity and linearity assumptions are often adequate,
yet can lead to poor performance when dealing with some practical applications.
In this paper, we model each variable in graph G as a polynomial regression of
its parents to capture complex relationships between individual variables and
with a utility function of polynomial form. We develop a message-passing
algorithm to propagate information throughout the network solely using moments
which enables the expected utility scores to be calculated exactly. Our
propagation method scales up well and enables to perform inference in terms of
a finite number of expectations. We illustrate how the proposed methodology
works with examples and in an application to decision problems in energy
planning and for real-time clinical decision support.
