The numeraire e-variable and reverse information projection

TL;DR

This paper introduces the numeraire e-variable as a universal, optimal test statistic for composite hypotheses, linking it to reverse information projection and providing tools for practical computation and broader utility interpretations.

Contribution

It establishes the existence of a universal numeraire e-variable that generalizes the reverse information projection without assumptions, connecting hypothesis testing, information geometry, and utility optimization.

Findings

Existence of a universal numeraire e-variable for any null and alternative.

Identification of the numeraire with the reverse information projection.

Application of the theory to nonparametric examples and utility-based interpretations.

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.