Information projections have found many important applications in probability theory, statistics, and related fields. In the field of hypothesis testing in particular, the reverse information projection (RIPr) has recently been shown to lead to so-called growth-rate optimal (GRO) e-statistics for testing simple alternatives against composite null hypotheses. However, the RIPr as well as the GRO criterion are only defined in cases where the infimum information divergence between the null and alternative is finite. Here, we show that under much weaker conditions there often still exists an element in the alternative that is `closest' to the null: the universal reverse information projection. The universal reverse information projection and its non-universal counterpart coincide whenever the KL is finite, and the strictness of this generalization will be shown by an example. Furthermore, the universal RIPr leads to optimal e-statistics in a sense that is a novel, but natural, extension of the GRO criterion. Finally, we discuss conditions under which the universal RIPr is a strict sub-probability distributions, and conditions under which an approximation of the universal RIPr leads to approximate e-statistics.

翻译：信息投影在概率论、统计学及相关领域中有许多重要应用。特别是在假设检验领域，逆向信息投影（RIPr）已被证明能够生成所谓的增长率最优（GRO）E统计量，用于检验简单备择假设与复合零假设的对立。然而，RIPr及GRO准则仅在零假设与备择假设之间的信息散度有限时才有定义。本文表明，在更弱的条件下，通常仍存在一个备择假设中的元素“最接近”零假设：即通用逆向信息投影。当KL散度有限时，通用逆向信息投影与其非通用版本重合，且将通过示例证明这一推广的严谨性。此外，通用RIPr在一种新颖但自然的GRO准则扩展意义上可生成最优E统计量。最后，我们讨论了通用RIPr成为严格子概率分布的条件，以及近似通用RIPr生成近似E统计量的条件。