Information projections have found important applications in probability theory, statistics, and related areas. 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 undefined whenever the infimum information divergence between the null and alternative is infinite. We show that in such scenarios 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 information divergence is finite. Furthermore, the universal RIPr is shown to lead to optimal e-statistics in a sense that is a novel, but natural, extension of the GRO criterion. We also give conditions under which the universal RIPr is a strict sub-probability distribution, as well as conditions under which an approximation of the universal RIPr leads to approximate e-statistics. For this case we provide tight relations between the corresponding approximation rates.

翻译：信息投影在概率论、统计学及相关领域已有重要应用。特别是在假设检验领域，最近研究表明，逆信息投影（RIPr）可导出检验简单备择假设与复合零假设的所谓增长速率最优（GRO）E统计量。然而，当零假设与备择假设之间的信息散度无穷大时，RIPr及GRO准则均无法定义。我们证明，在此类情况下往往仍存在备择假设中"最接近"零假设的元素——通用逆信息投影。在信息散度有限时，通用逆信息投影与其非通用形式一致。此外，通用RIPr能以新颖但自然的GRO准则扩展形式导出最优E统计量。我们给出了通用RIPr为严格子概率分布的条件，以及通用RIPr近似导致近似E统计量的条件，并针对后者提供了对应逼近速率间的紧致关系。