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Entropic Inequalities and Marginal Problems

机译：熵不等式和边际问题

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摘要

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 nontrivial 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.
机译：一个边际问题问，对于某些随机变量集，给定的边际分布族是否源自这些变量的某种联合分布。在这里，我们指出，这种联合分布的存在已经在给定边际的香农熵水平上施加了非平凡条件。这些熵不等式是存在联合分布的必要（但不充分）标准。对于每个边际问题，可以通过傅立叶–莫兹金消除法来计算此类香农型熵不等式的列表，并且我们为此提供了与傅立叶–莫兹金求解器的软件接口。对于给定边际的超图是一个循环图的情况，我们为分类所有相关熵不等式的问题提供了一个完整的解析解决方案，并使用此结果来限制随机过程中相关性的衰减。此外，我们表明，微分熵的香农型不等式与连续变量边际问题无关。非香农类型的不等式在离散情况和连续情况下都存在。与其他方法相比，我们的通用框架很容易适应这样的情况，例如在图形模型的情况下，对联合分布有其他（条件）独立性要求。我们以未解决的问题列表结束。补充文章讨论了量子非局部性和上下文性的应用。

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