使用Blackwell-Cover猜测策略分离未观测的伯努利试验

中文摘要 AI 辅助

Blackwell和Cover都提出了相同的方法，用于在只能观测到一对数字中的一个而另一个保持未观测时，猜测这对数字中较大的一个。按照最初的表述，该过程以一个计算结束，该计算表明正确猜测较大数字的概率大于二分之一。在本文中，我们研究当采用Blackwell/Cover过程时，关于这对数字中未被观测的那个数字，我们能得出什么结论。设α1和α2是(0,1]中的两个不同数字。设G是[0, α1]上的均匀分布，H是[0, α2]上的均匀分布。我们将G和H视为由硬币的正面概率所表示。我们使用Blackwell-Cover策略的一种变体，从由G和H创建的分布中选取成对的硬币，使得其中一枚硬币被翻转，并且由于该次翻转，未被观测（未翻转）的硬币被放入两个集合A和B之一。我们证明，在某些情况下，可以做到使得A中未观测硬币的正面概率的均值小于B中未观测硬币的正面概率的均值。我们证明，在某些情况下，递归应用此过程会导致一个集合几乎完全由来自两个分布G和H之一的硬币组成，而另一个集合几乎完全由来自另一个分布的硬币组成。

英文摘要

Blackwell and Cover both proposed the same method for guessing the larger of a pair of two numbers when one can only observe one and the other of the two numbers remains unobserved. As originally stated, the procedure ended with a computation showing that the probability of correctly guessing the larger number was greater than one-half. In this paper, we investigate what we can say when about the number in the pair that was not observed we employ the Blackwell/Cover procedure. Let α1 and α2 be two different numbers in (0,1]. Let G be the uniform distribution on [0, α1] and H be the uniform distribution on [0, α2]. We think of G and H as being represented by the heads probabilities of coins. We use a variant of the Blackwell-Cover strategy ro take pairs of coins chosen from distributions created from G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. We show that there are situations in which recursive application of this process results in a set consisting almost completely of coins from one of the two distributions G and H, and another set consisting almost completely of coins from the other distribution.

发表机构

• California State University (Long Beach)（加州州立大学（长滩））

机构由 AI 辅助整理，请以论文原文为准。

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17 pages