@MISC{Sears_ahammersley-clifford, author = {T. Sears and P. Sunehag}, title = {A Hammersley-Clifford Thorem for Induced Graph Semantics}, year = {} }

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Abstract

The Hammersley-Clifford Theorem provides a key link which makes graphical models a useful way to represent certain assumptions employed in statistical models. The theorem links the absence of edges in an undirected graph to a Markov assumption about the involved random variables, which are taken to be the nodes of the graph. The theorem states that a density satisfies certain conditional independence assumptions if and only if it factorizes according to the clique structure of the graph. Sometimes this result is taken to mean that the exponential family is the only distribution type suitable for use with graphical models, since its factorization properties are compatible with the theorem. In this paper we generalize the Hammersley-Clifford Theorem to q-exponential families. These distributions arise in the study of Tsallis entropy. Important properties of distributions related to Tsallis entropy include the possibility of generating distributions with power-law behavior and distributions with finite support. We achieve the generalization by introducing a variation on the Markov property, thereby inducing a new meaning for the edge of a graph. Although the graph is always fully connected in the original semantics, our theorem implies that the computational requirements of inference are not as severe as this might suggest. We also provide further generalization to φ-exponential families which arise in connection to entropies defined using deformed logarithms. References: [1] Murray Gell-Mann and Constantino Tsallis. Nonextensive Entropy, Sante Fe