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 transformation of the latent variables taking arbitrary distributions. We use Gaussian models and independent factor models as examples to demonstrate the use of CFGs. 
 
The requirement of a linear transformation between latent variables (with certain independence restriction) and the observed 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 MFG with identical structure. This Fourier transform duality allows inference problems on a CFG to be solved on the corresponding dual MFG.