bibkey: wainwrightjordan2008graphical authors: Martin J. Wainwright and Michael I. Jordan year: 2008 title: Graphical Models, Exponential Families, and Variational Inference doi: 10.1561/2200000001 url: https://people.eecs.berkeley.edu/~jordan/papers/wainwright-jordan-fnt.pdf claim: “Junction-tree marginals factor through their shared separators; the paper defines mutual information and Kullback–Leibler divergence by probability-ratio logarithms.” strata_touched: [] license: citation-only triage: anchor
Shared separators and finite probability-ratio information
The author PDF is the Foundations and Trends in Machine Learning 1(1–2), 1–305 text. Section 2.5.2, equation (2.12), printed p. 32, divides products of clique marginals by separator marginals with the indicated multiplicities. Example 2.2, pp. 32–33, specializes to a three-node Markov chain:
Proposition 2.1 on p. 33 gives a distribution realizing normalized, locally consistent junction-tree marginals. This constructs a completion; it does not identify every arbitrary joint law with those marginals. The pyramid reference, §六 uses the two-clique instance with separator . At zero separator mass all associated nonnegative cells vanish; the consumer assigns zero there and proves normalization without dividing by zero.
Equation (4.13), p. 82, defines pairwise mutual information by the joint-to-product ratio. Section 5.2.2, equation (5.9), p. 132, defines the modern Kullback–Leibler divergence as the expectation under of . The consumer’s §九 uses base-two finite sums, states support inclusion and proves nonnegativity by the ordinary log inequality, including boundary probabilities. Its exact entropy-gap identity, native reply loss and degenerate apex treatment are the consumer’s elementary finite-law applications. They are not attributed to the paper’s exponential-family parameter formulas or to a missing primary historical source.