Hammersley-Clifford¶
A positive distribution \(p(y) > 0\) satisfies the CI properties of an undirected graph G iff p can be represented as a product of factors, one per maximal clique, i.e.,
\[p(y|\theta) = \frac{1}{Z(\theta)} \prod_{c \in C} \psi_c(y_c|\theta) \]
where C is the set of all the (maximal) cliques of G and \(Z(\theta)\) is the partition function given by
\[Z(\theta) \triangleq \sum_x \prod_{c \in C} \psi_c (y_c|\theta_c) \]
this partition function is what ensures the overall distribution sums to 1
Example
$\( p(y|\theta) = \frac{1}{Z(\theta)} \psi_{123}(y_1, y_2, y_3)\psi_{234}(y_2, y_3, y_4) \phi_{35}(y_3,y_5) \\ Z(\theta) = \sum_y \psi_{123}(y_1, y_2, y_3)\psi_{234}(y_2, y_3, y_4) \phi_{35}(y_3,y_5)\)$