Loopy Belief Propogation and Gibbs Measures

We address the question of convergence in the loopy belief propagation (LBP) algorithm. Specifically, we relate convergence of LBP to the existence of a weak limit for a sequence of Gibbs measures defined on the LBP s associated computation tree.Using tools FROM the theory OF Gibbs measures we develop easily testable sufficient conditions FOR convergence.The failure OF convergence OF LBP implies the existence OF multiple phases FOR the associated Gibbs specification.These results give new insight INTO the mechanics OF the algorithm.

💡 Research Summary

The paper investigates the convergence properties of the loopy belief propagation (LBP) algorithm by establishing a rigorous connection with the theory of Gibbs measures. LBP is a message‑passing scheme used to approximate marginal distributions in graphical models such as Bayesian networks and Markov random fields. While it is exact on trees, its behavior on graphs containing cycles (the “loopy” case) is notoriously unpredictable: the algorithm may converge to a fixed point, oscillate, or diverge. Existing convergence analyses typically focus on the dynamics of the message updates (e.g., contraction mappings, spectral properties) but do not directly address the underlying global probability structure.

The authors adopt a different viewpoint: each iteration of LBP can be interpreted as expanding a computation tree that unrolls the original graph. For a given depth (n), the computation tree (T_n) exactly reproduces the local neighborhood of the original graph up to (n) steps. The sequence of messages generated by LBP therefore defines a sequence of probability measures ({\mu_n^{(t)}}) on these finite trees. As (t) (the iteration index) grows, the measures evolve according to the local conditional specifications of the graphical model.

In the language of statistical physics, a Gibbs specification (\gamma) assigns to each finite subset (\Lambda) a conditional distribution (\gamma_\Lambda(\cdot\mid \sigma_{\Lambda^c})). A global Gibbs measure (\mu) is a probability measure on the infinite configuration space that is consistent with (\gamma) on every finite (\Lambda). The crucial notion is that of a weak limit: the measures (\mu_n^{(t)}) converge weakly if, for every finite set of variables, the marginal distributions stabilize as (t\to\infty). The central theorem of the paper states that:

LBP converges ⇔ the sequence of Gibbs measures on the computation tree has a weak limit ⇔ the associated infinite‑volume Gibbs measure is unique.

The proof proceeds in three steps. First, if the messages converge, the induced marginals on any finite subtree converge, yielding a consistent family of limit marginals. By Kolmogorov’s extension theorem this family defines a unique infinite‑volume Gibbs measure. Second, uniqueness of the Gibbs measure forces the conditional specifications to be continuous with respect to boundary conditions, which in turn guarantees that the message updates are a contraction and thus converge. Finally, if the messages do not converge, the lack of a weak limit implies the existence of multiple Gibbs measures (multiple phases) for the same local specification.

To make the uniqueness condition operational, the authors invoke the classic Dobrushin condition from Gibbs measure theory. For each variable (i) they define a variation coefficient \