Factorization of Discrete Probability Distributions

We formulate necessary and sufficient conditions for an arbitrary discrete probability distribution to factor according to an undirected graphical model, or a log-linear model, or other more general exponential models. This result generalizes the well known Hammersley-Clifford Theorem.

💡 Research Summary

The paper establishes a comprehensive set of necessary and sufficient conditions under which an arbitrary discrete probability distribution can be factorized according to an undirected graphical model, a log‑linear model, or more generally any exponential family model. It begins by revisiting the classic Hammersley‑Clifford theorem, which guarantees factorization for strictly positive distributions that satisfy the global Markov property. Recognizing that the positivity assumption is often violated in real‑world data—especially in sparse or constrained settings—the authors introduce a new concept called “S‑Markovness.” Here S denotes the support of the distribution, and S‑Markovness requires that any pair of non‑adjacent variables be conditionally independent when restricted to S. This notion decouples the graphical structure from the global positivity requirement and allows the authors to prove that any distribution satisfying S‑Markovness can be expressed as a log‑linear model:

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