2 research outputs found
Entropic Inequalities and Marginal Problems
A marginal problem asks whether a given family of marginal distributions for
some set of random variables arises from some joint distribution of these
variables. Here we point out that the existence of such a joint distribution
imposes non-trivial conditions already on the level of Shannon entropies of the
given marginals. These entropic inequalities are necessary (but not sufficient)
criteria for the existence of a joint distribution. For every marginal problem,
a list of such 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 graph, we provide a complete analytic solution to the
problem of classifying all relevant entropic inequalities, and use this result
to bound the decay of correlations in stochastic processes. Furthermore, we
show that Shannon-type inequalities for differential entropies are not relevant
for continuous-variable marginal problems; non-Shannon-type inequalities are,
both in the discrete and in the continuous case. In contrast to other
approaches, 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 complementary article discusses applications to quantum nonlocality and
contextuality.Comment: 26 pages, 3 figure