论文标题
“选择最佳D样品”模型中的网格熵
Grid entropy in a "choose the best of D samples" model
论文作者
论文摘要
网格熵是晶格模型固有的确定性数量，它沿沿路径的经验度量的熵微弱地收敛到给定的目标度量。在本文中，我们研究了一个模型中经验措施的限制行为，该模型包括反复从分布中获取$ d $样本，并根据无所不知的“策略”挑选出来。我们表明，经验措施的一组限制点几乎肯定是相同的，无论我们是否将自己限制为独立于所有过去和未来选择的策略，以及此外，这一组合与有限的网格熵相吻合。该集合是凸面和弱紧凑的。我们将其极端点描述为由自然的“贪婪”确定性策略所给出的，并计算上述极端点的网格熵为0。这使得对经验度量的限制点的描述表示为封闭的密度凸出的凸壳，该密度是$ d \ cdot beta（1，d）$分布的密度。我们还得出了该模型的Gibbs自由能的基于网格熵的变化公式的简化版本，并介绍了网格熵的双公式。
Grid entropy is a deterministic quantity inherent to lattice models which captures the entropy of empirical measures along paths that converge weakly to a given target measure. In this paper, we study the limiting behaviour of empirical measures in a model consisting of repeatedly taking $D$ samples from a distribution and picking out one according to an omniscient "strategy." We show that the set of limit points of empirical measures is almost surely the same whether or not we restrict ourselves to strategies which make the choices independently of all past and future choices, and furthermore, that this set coincides with the set of measures with finite grid entropy. This set is convex and weakly compact; we characterize its extreme points as those given by a natural "greedy" deterministic strategy and we compute the grid entropy of said extreme points to be 0. This yields a description of the set of limit points of empirical measures as the closed convex hull of measures given by a density which is $D \cdot Beta(1,D)$ distributed. We also derive a simplified version of a grid entropy-based variational formula for Gibbs Free Energy for this model, and we present the dual formula for grid entropy.
