We consider the problem of finding maximum values of f-divergences D_f(P ∥ Q) of discrete probability distributions P and Q with values on a finite set under the condition that the variation distance V(P, Q) between them and one of the distributions P or Q are given. We obtain exact expressions for such maxima of f-divergences, which in a number of cases allow to obtain both explicit formulas and upper bounds for them. As a consequence, we obtain explicit expressions for the maxima of f-divergences D_f (P ∥ Q) given that, besides V(P, Q), we only know the value of the maximum component of either P or Q. Analogous results are also obtained for the Rényi divergence.

中文翻译：

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给定变分距离下f-散度和Rényi发散的最大值

我们考虑寻找的最大值的问题˚F -divergences d _˚F（P离散概率分布的∥Q）P和Q与条件下的有限集合的值，该偏差距离V（P，Q他们中的一个之间）分布P或Q中给出。我们获得了此类f散度最大值的精确表达式，在许多情况下，这些表达式允许获得其明确的公式和上限。结果，我们获得了f-散度D _f（P ∥ Q因为，除了）V（P，Q），我们只知道无论是中最大成分的值P或Q。Rényi散度也获得了类似的结果。