We consider the problem of finding the maximum values of divergences D(P‖Q) and D(Q‖P) for probability distributions P and Q ranging in the finite set \(\mathcal{N}=\left\{1,\;2,...,n\right\}\) provided that both the variation distance V (P,Q) between them and either the probability distribution Q or (in the case of D(P‖Q)) only the value of the minimal component q_min of the probability distribution Q are given. Precise expressions for the maximum values of these divergences are obtained. In several cases these expressions allow us to write out some explicit formulas and simple upper and lower bounds for them. Moreover, explicit formulas for the maximum of D(P‖Q) for given V (P,Q) and q_min and also for the maximum of D(Q‖P) for given Q and V (P,Q) are obtained for all possible values of these parameters.

中文翻译：

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给定变分距离下有限维分布发散的最佳上限

我们考虑寻找分歧的最大值的问题d（P ‖ Q）和d（Q ‖ P）为概率分布P和Q的范围在有限集合\（\ mathcal {N} = \左\ {1，\ ; 2，...，N \右\} \）提供的是，偏差距离V（P，Q它们，然后要么的概率分布之间）Q或者（在的情况下，d（P ‖ Q））只值分布Q的最小分量q _min给出。获得了这些差异最大值的精确表达式。在某些情况下，这些表达式使我们能够为它们写出一些明确的公式以及简单的上下限。此外，对于最大的明确的公式d（P ‖ Q）对于给定的V（P，Q）和q_分钟，也为最大的d（Q ‖ P为给出）Q和V（P，Q被用于获得）这些参数的所有可能值。