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Let Xi, i ∊ V form a Markov random field (MRF) represented by an undirected graph G = (V, E), and V' be a subset of V. We determine the smallest graph that can always represent the subfield Xi, i ∊ V' as an MRF. Based on this result, we obtain a necessary and sufficient condition for a subfield of a Markov tree to be also a Markov tree. When G is a path so that Xi, i ∊ V form a Markov chain, it is...
Let $ \mathcal {N}=\{1,\ldots ,n\}$ . The entropy function h of a set of $n$ discrete random variables $\{X_{i}:i\in \mathcal {N} \}$ is a $2^{n}$ -dimensional vector whose entries are $ \text {h}( \mathcal {A})\triangleq H(X_{ \mathcal {A}})$ , $ \mathcal {A}\subset \mathcal {N} $ , the (joint) entropies of the subsets of the set of $n$ random variables with $H(X_\emptyset )=0$ by convention...
This paper investigates when Shannon-type inequalities completely characterize the part of the closure of the entropy region Γ̅n* that is symmetric under the action of a specified random variable permutation group. This question is answered exhaustively for every group permuting n = 4 and n = 5 random variables, while multiple examples for arbitrary n ≥ 6 are provided for both tightness and non-tightness...
We prove that by imposing a conditional mutual independence constraint and a marginalisation constraint, the almost entropic region can be completely characterised by Shannon-type information inequalities. Such a property is applied to obtain an explicit lower bound on the generalised Wyner common information.
Contrary to the traditional method of information inequalities, in this paper, the entropy function region Γn* and its closure ̄Γn* are characterized via extreme rays of its outer bound polymatroidal region Γn. The characterization of Γ3* and the tightness of Γn as an outer bound on ̄Γn* are studied.
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