The numeraire e-variable and reverse information projection

Abstract: We consider testing a composite null hypothesis $\mathcal{P}$ against a point alternative $\mathsf{Q}$ using e-variables, which are nonnegative random variables $X$ such that $\mathbb{E}\mathsf{P}[X] \leq 1$ for every $\mathsf{P} \in \mathcal{P}$. This paper establishes a fundamental result: under no conditions whatsoever on $\mathcal{P}$ or $\mathsf{Q}$, there exists a special e-variable $X^*$ that we call the numeraire, which is strictly positive and satisfies $\mathbb{E}\mathsf{Q}[X/X^*] \leq 1$ for every other e-variable $X$. In particular, $X^*$ is log-optimal in the sense that $\mathbb{E}_\mathsf{Q}[\log(X/X^*)] \leq 0$. Moreover, $X^*$ 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 and RIPr in concrete cases. We discuss several nonparametric examples where we can indeed identify the numeraire and RIPr, despite not having a reference measure. Our results have interpretations outside of testing in that they yield the optimal Kelly bet against $\mathcal{P}$ if we believe reality follows $\mathsf{Q}$. 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.