In undirected graphical models (Markov random fields) the joint distribution is written as a normalized product of nonnegative potential functions, one per clique of the graph, divided by a partition function that sums or integrates the unnormalized product over all configurations — a sum that is exponential in the number of variables and is the principal source of computational difficulty, since unlike directed models the factors are not conditional distributions and carry no probabilistic interpretation of their own. The Hammersley–Clifford theorem states that a strictly positive probability distribution factorizes over the cliques of a graph if and only if it can be written as a product of exponentials of clique energy functions, which converts the unconstrained nonnegativity requirement on potentials into an unconstrained real-valued energy and links high-probability configurations to low energy in the manner of statistical physics. The Markov property that names the model is that each variable is conditionally independent of all others given its immediate neighbours (the undirected analogue of conditioning on the immediate predecessor in a Markov chain); exact inference over such a model is tractable only when the graph is a tree, and loops force approximate inference.
In undirected graphical models (Markov random fields) the joint distribution is written as a normalized product of nonnegative potential functions, one per clique of the graph, divided by a partition…