We consider testing a composite null hypothesis $\mathcal{P}$ against a point alternative $\mathsf{Q}$. This paper establishes a powerful and general result: under no conditions whatsoever on $\mathcal{P}$ or $\mathsf{Q}$, there exists a special e-variable $X^*$ that we call the numeraire. It is strictly positive and for every $\mathsf{P} \in \mathcal{P}$, $\mathbb{E}_\mathsf{P}[X^*] \le 1$ (the e-variable property), while for every other e-variable $X$, we have $\mathbb{E}_\mathsf{Q}[X/X^*] \le 1$ (the numeraire property). In particular, this implies $\mathbb{E}_\mathsf{Q}[\log(X/X^*)] \le 0$ (log-optimality). $X^*$ also identifies a particular sub-probability measure $\mathsf{P}^*$ via the density $d \mathsf{P}^*/d \mathsf{Q} = 1/X^*$. As a result, $X^*$ can be seen as a generalized likelihood ratio of $\mathsf{Q}$ against $\mathcal{P}$. We show that $\mathsf{P}^*$ coincides with the reverse information projection (RIPr) when additional assumptions are made that are required for the latter to exist. Thus $\mathsf{P}^*$ is a natural definition of the RIPr in the absence of any assumptions on $\mathcal{P}$ or $\mathsf{Q}$. In addition to the abstract theory, we provide several tools for finding the numeraire in concrete cases. We discuss several nonparametric examples where we can indeed identify the numeraire, despite not having a reference measure. We end with a more general optimality theory that goes beyond the ubiquitous logarithmic utility. We focus on certain power utilities, leading to reverse R\'enyi projections in place of the RIPr, which also always exist.

翻译：我们考虑针对点备择假设$\mathsf{Q}$检验复合零假设$\mathcal{P}$。本文建立了一个强大且通用的结论：无论对$\mathcal{P}$或$\mathsf{Q}$施加何种条件，总存在一个特殊的e变量$X^*$，我们称之为计价单位。该变量严格为正，且对每个$\mathsf{P} \in \mathcal{P}$满足$\mathbb{E}_\mathsf{P}[X^*] \le 1$（e变量性质），同时对任意其他e变量$X$均有$\mathbb{E}_\mathsf{Q}[X/X^*] \le 1$（计价单位性质）。特别地，这意味着$\mathbb{E}_\mathsf{Q}[\log(X/X^*)] \le 0$（对数最优性）。$X^*$还通过密度$d \mathsf{P}^*/d \mathsf{Q} = 1/X^*$确定了一个特定的子概率测度$\mathsf{P}^*$。因此，$X^*$可视为$\mathsf{Q}$相对于$\mathcal{P}$的广义似然比。我们证明，当对反向信息投影（RIPr）存在性施加额外假设时，$\mathsf{P}^*$与反向信息投影一致。因此，在$\mathcal{P}$或$\mathsf{Q}$无任何假设的条件下，$\mathsf{P}^*$是反向信息投影的自然定义。除抽象理论外，我们还提供了若干具体情形下寻找计价单位的工具。我们讨论了多个非参数示例，尽管缺乏参考测度，仍能成功识别计价单位。最后，我们拓展了超越泛用对数效用的更一般最优性理论，重点研究特定幂效用函数，由此导出始终存在的反向Rényi投影以取代反向信息投影。