Bleuler Cédric, Lapidoth Amos, Pfister Christoph
Signal and Information Processing Laboratory, ETH Zurich, 8092 Zurich, Switzerland.
Entropy (Basel). 2020 Mar 11;22(3):316. doi: 10.3390/e22030316.
Motivated by a horse betting problem, a new conditional Rényi divergence is introduced. It is compared with the conditional Rényi divergences that appear in the definitions of the dependence measures by Csiszár and Sibson, and the properties of all three are studied with emphasis on their behavior under data processing. In the same way that Csiszár's and Sibson's conditional divergence lead to the respective dependence measures, so does the new conditional divergence lead to the Lapidoth-Pfister mutual information. Moreover, the new conditional divergence is also related to the Arimoto-Rényi conditional entropy and to Arimoto's measure of dependence. In the second part of the paper, the horse betting problem is analyzed where, instead of Kelly's expected log-wealth criterion, a more general family of power-mean utility functions is considered. The key role in the analysis is played by the Rényi divergence, and in the setting where the gambler has access to side information, the new conditional Rényi divergence is key. The setting with side information also provides another operational meaning to the Lapidoth-Pfister mutual information. Finally, a universal strategy for independent and identically distributed races is presented that-without knowing the winning probabilities or the parameter of the utility function-asymptotically maximizes the gambler's utility function.
受一个赛马投注问题的启发，引入了一种新的条件雷尼散度。将其与在Csiszár和Sibson的相依性度量定义中出现的条件雷尼散度进行比较，并研究了这三种散度的性质，重点是它们在数据处理下的行为。与Csiszár和Sibson的条件散度分别引出各自的相依性度量的方式相同，新的条件散度也引出了拉皮多斯 - 普菲斯特互信息。此外，新的条件散度还与有里本 - 雷尼条件熵以及有里本的相依性度量相关。在论文的第二部分，分析了赛马投注问题，其中考虑的不是凯利的预期对数财富准则，而是一个更一般的幂平均效用函数族。分析中的关键作用由雷尼散度发挥，并且在赌徒可以获取边信息的设定中，新的条件雷尼散度是关键。有边信息的设定也为拉皮多斯 - 普菲斯特互信息提供了另一种操作意义。最后，提出了一种针对独立同分布赛事的通用策略，该策略在不知道获胜概率或效用函数参数的情况下，渐近地最大化赌徒的效用函数。
