We consider the problem of finding the minimum and maximum values of f-divergence for discrete probability distributions P and Q provided that one of these distributions and the value of their coupling are given. An explicit formula for the minimum value of the f-divergence under the above conditions is obtained, as well as a precise expression for its maximum value. This precise expression is not explicit in the general case, but in many special cases it allows us to write out both explicit formulas and simple upper bounds, which are sometimes optimal. Similar explicit formulas and upper bounds are also obtained for the Kullback–Leibler and χ² divergences, which are the most important cases of the f-divergence.

我们考虑找到离散概率分布P和Q的f-散度的最小值和最大值的问题，只要给出这些分布之一及其耦合值即可。获得了在上述条件下f-散度最小值的明确公式，以及其最大值的精确表达式。这种精确的表达式在一般情况下不是明确的，但是在许多特殊情况下，它使我们既可以写出明确的公式，又可以写出有时是最优的简单上限。类似明确的公式和上界也获得了库勒巴克-莱布勒和χ ^2个分歧，哪些是最重要的案件f-散度。