Convolutional Factor Graphs as Probabilistic Models
  Based on a recent development in the area of error control coding, we
introduce the notion of convolutional factor graphs (CFGs) as a new class of
probabilistic graphical models. In this context, the conventional factor graphs
are referred to as multiplicative factor graphs (MFGs). This paper shows that
CFGs are natural models for probability functions when summation of independent
latent random variables is involved. In particular, CFGs capture a large class
of linear models, where the linearity is in the sense that the observed
variables are obtained as a linear ransformation of the latent variables taking
arbitrary distributions. We use Gaussian models and independent factor models
as examples to emonstrate the use of CFGs. The requirement of a linear
transformation between latent variables (with certain independence restriction)
and the bserved variables, to an extent, limits the modelling flexibility of
CFGs. This structural restriction however provides a powerful analytic tool to
the framework of CFGs; that is, upon taking the Fourier transform of the
function represented by the CFG, the resulting function is represented by a FG
with identical structure. This Fourier transform duality allows inference
problems on a CFG to be solved on the corresponding dual MFG.
