Computer Science – Information Theory
Scientific paper
2011-12-20
Computer Science
Information Theory
20 pages, 2 figures
Scientific paper
The marginal problem asks when a given family of marginal distributions for some set of random variables can be extended to a joint distribution of these variables. Here we point out that the existence of a joint distribution imposes non-trivial conditions already on the level of Shannon entropies of the given marginals. For every marginal problem, a list of such conditions in terms of Shannon-type entropic inequalities can be calculated by Fourier-Motzkin elimination, and we offer a software interface to a Fourier-Motzkin solver for doing so. For the case that the hypergraph of given marginals is a cycle, we provide a complete analytic solution to the problem of classifying all tight entropic inequalities, and use this result to obtain a bound on the decay of correlations in stochastic processes. We show that Shannon-type inequalities for differential entropies are not relevant for the continuous-variable marginal problem; non-Shannon-type inequalities are, both in the discrete and in the continuous case. Our general framework easily adapts to situations where one has additional (conditional) independence requirements on the joint distribution, as in the case of graphical models. We end with a list of open problems. A forthcoming article will discuss applications to quantum nonlocality and contextuality.
Chaves Rafael
Fritz Tobias
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