Factor graph

A factor graph is an $X,F$-bipartite graph where $X=\{X_{1},X_{2},\dots ,X_{n}\}$ is a set of variables and $F=\{f_{1},f_{2},\dots ,f_{m}\}$ is a set of factors. A factor $f_{j}$ is a non-negative function, usually between 0 and 1, mapping from a subset of variables $X_{j}\subseteq X$. This graph represents the factorisation

$g(\mathbf {x} )=\prod _{j=1}^{m}f_{j}(\mathbf {x_{j}} ),$

where $\mathbf {x}$ is an assignment to all values of $X$ and $\mathbf {x_{j}}$ is an assignment to all values of $X_{j}$.