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Mutual Information Bounds via Adjacency Events

机译：通过邻接事件的相互信息界限

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摘要

The mutual information between two jointly distributed random variables X and Y is a functional of the joint distribution PXY, which is sometimes difficult to handle or estimate. A coarser description of the statistical behavior of (X, Y) is given by the marginal distributions PX, PY and the adjacency relation induced by the joint distribution, where x and y are adjacent if P(x, y) > 0. We derive a lower bound on the mutual information in terms of these entities. The bound is obtained by viewing the channel from X to Y as a probability distribution on a set of possible actions, where an action determines the output for any possible input, and is independently drawn. We also provide an alternative proof based on convex optimization that yields a generally tighter bound. Finally, we derive an upper bound on the mutual information in terms of adjacency events between the action and the pair (X, Y), where in this case, an action a and a pair (x, y) are adjacent if y = a(x). As an example, we apply our bounds to the binary deletion channel and show that for the special case of an independent identically distributed input distribution and a range of deletion probabilities, our lower and upper bounds both outperform the best known bounds for the mutual information.
机译：两个联合分布的随机变量X和Y之间的互信息是联合分布PXY的函数，有时难以处理或估计。 （X，Y）的统计行为的粗略描述由边际分布PX，PY和联合分布引起的邻接关系给出，如果P（x，y）> 0，则x和y相邻。我们得出就这些实体而言，相互信息的下限。通过将从X到Y的通道视为一组可能动作的概率分布来获得边界，其中动作确定任何可能输入的输出，并独立绘制。我们还提供了基于凸优化的替代证明，该证明通常会产生更严格的界限。最后，我们根据动作与对（X，Y）之间的邻接事件得出互信息的上限，在这种情况下，如果y = a，则动作a和一对（x，y）是相邻的（X）。例如，我们将边界应用于二进制删除通道，并表明对于独立的相同分布的输入分布和一定范围的删除概率的特殊情况，我们的下限和上限均优于已知的互信息范围。

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