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Explicit Bounds for Entropy Concentration Under Linear Constraints

机译：线性约束下熵集中的显式界

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摘要

Consider the set of all sequences of outcomes, each taking one of values, whose frequency vectors satisfy a set of linear constraints. If is fixed while increases, most sequences that satisfy the constraints result in frequency vectors whose entropy approaches that of the maximum entropy vector satisfying the constraints. This well-known entropy concentration phenomenon underlies the maximum entropy method. Existing proofs of the concentration phenomenon are based on limits or asymptotics and unrealistically assume that constraints hold precisely, supporting maximum entropy inference more in principle than in practice. We present, for the first time, non-asymptotic, explicit lower bounds on for a number of variants of the concentration result to hold to any prescribed accuracies, with the constraints holding up to any specified tolerance, considering the fact that allocations of discrete units can satisfy constraints only approximately. Again unlike earlier results, we measure concentration not by deviation from the maximum entropy value, but by the and distances from the maximum entropy-achieving frequency vector. One of our results holds independently of the alphabet size and is based on a novel proof technique using the multi-dimensional Berry–Esseen theorem. We illustrate and compare our results using various detailed examples.
机译：考虑所有结果序列的集合，每个序列取一个值，其频率向量满足一组线性约束。如果在增加的同时固定，则大多数满足约束条件的序列会导致频率向量的熵接近满足约束条件的最大熵向量的熵。这种众所周知的熵集中现象是最大熵方法的基础。集中现象的现有证据是基于极限或渐近性的，并且不切实际地假设约束条件精确地成立，因此在理论上比在实践中更多地支持最大熵推断。考虑到离散单元的分配这一事实，我们首次提出了非渐近的，显着的下界，其浓度结果的许多变体可以保持任何规定的精度，并且约束条件可以保持任何指定的公差。只能大致满足约束条件再次不同于先前的结果，我们不是通过偏离最大熵值来测量浓度，而是通过与达到最大熵的频率向量的和来测量浓度。我们的结果之一与字母大小无关，并且基于使用多维Berry-Esseen定理的新颖证明技术。我们使用各种详细的示例来说明和比较我们的结果。

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