The numeraire e-variable and reverse information projection

TL;DR

The paper develops a universal theory for e-variables in testing composite nulls against a point alternative, proving the existence and uniqueness of a numeraire e-variable $X^*$ that is strictly positive and log-optimal under ${\\mathsf Q}$. It shows that $X^*$ induces a reverse information projection ${\\mathsf P}^*$ via $d{\\mathsf P}^*/d{\\mathsf Q}=1/{\\mathbf X}^*$, and establishes a strong duality between maximizing ${\\mathbb E}_{\\mathsf Q}[\\log X]$ over e-variables and minimizing ${\\mathsf E}_{\\mathsf Q}[\\log(d{\\mathsf P}^a/d{\\mathsf Q})]$ over the effective null ${\\mathcal P}_{\\textnormal{eff}}$, with $X^*$ and ${\\mathsf P}^*$ coinciding with RIPr under additional assumptions. The work provides practical tools to identify the numeraire and RIPr in nonparametric settings without a reference measure, and demonstrates several concrete examples including bounded-mean, sub-Gaussian, symmetric, and one-dimensional exponential family cases. It also generalizes the log-based framework to reverse Rényi projections for power utilities and discusses the inadmissibility of universal inference. The results offer a robust, assumption-free foundation for optimal evidence against complex nulls and have implications for Bayesian-like likelihood ratios and Kelly betting 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 Rényi projections in place of the RIPr, which also always exist.

Theorems & Definitions (64)

• Definition 2.1
• Example 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Remark 2.5
• Theorem 2.6
• proof
• Lemma 2.7