Shared information is a measure of mutual dependence among multiple jointly distributed random variables with finite alphabets. For a Markov chain on a tree with a given joint distribution, we give a new proof of an explicit characterization of shared information. The Markov chain on a tree is shown to possess a global Markov property based on graph separation; this property plays a key role in our proofs. When the underlying joint distribution is not known, we exploit the special form of this characterization to provide a multiarmed bandit algorithm for estimating shared information, and analyze its error performance.

翻译：共享信息是度量多个有限字母表联合分布随机变量间相互依赖关系的指标。针对树形结构上给定联合分布的马尔可夫链，本文给出了共享信息显式表征的新证明。研究表明，基于图分离特性的树状马尔可夫链具有全局马尔可夫性质，该性质在证明过程中起关键作用。当底层联合分布未知时，我们利用该表征的特殊形式设计了一种多臂赌博机算法以估计共享信息，并分析了其误差性能。