E-Values for Exponential Families: the General Case

TL;DR

This paper investigates optimal e-variables and e-processes for composite exponential family nulls, characterizing their properties, interrelations, and e-power, with a focus on Gaussian cases and high-dimensional settings.

Contribution

It introduces the reverse information projection (RIPr) for e-variables, analyzes their relationships with other methods, and quantifies e-power differences in high-dimensional scenarios.

Findings

RIPr prior characterized for Gaussian nulls and alternatives

COND e-variable often matches RIPr exactly or approximately

UI e-process has smaller e-power than COND by a dimension-dependent term

Abstract

We analyze common types of e-variables and e-processes for composite exponential family nulls: the optimal e-variable based on the reverse information projection (RIPr), the conditional (COND) e-variable, and the universal inference (UI) and sequen\-tialized RIPr e-processes. We characterize the RIPr prior for simple and Bayes-mixture based alternatives, either precisely (for Gaussian nulls and alternatives) or in an approximate sense (general exponential families). We provide conditions under which the RIPr e-variable is (again exactly vs. approximately) equal to the COND e-variable. Based on these and other interrelations which we establish, we determine the e-power of the four e-statistics as a function of sample size, exactly for Gaussian and up to $o(1)$ in general. For $d$-dimensional null and alternative, the e-power of UI tends to be smaller by a term of $(d/2) \log n + O(1)$ than that of the COND e-variable, which is the clear winner.