Hammersley-Clifford theorem

The Hammersley-Clifford theorem establishes that for strictly positive probability distributions, the conditional independence structure encoded by an undirected graph is exactly equivalent to a factorization of the distribution into a product of potential functions over the graph's maximal cliques. It is the undirected counterpart to the Causal Markov condition: where the Markov condition links directed graphs to probability through parenthood, Hammersley-Clifford links undirected graphs to probability through neighborhood. The theorem explains why Markov random fields work — and why they only work when the distribution is strictly positive, a condition that excludes many real-world systems with hard constraints, deterministic relationships, or structural zeros.

The positivity requirement is not merely technical. It reflects a deep assumption that the world is fundamentally stochastic rather than constrained, an assumption that fails in systems governed by conservation laws, logical necessities, or threshold effects.