# E-Values for Exponential Families: the General Case

arXiv: https://arxiv.org/abs/2409.11134 · HTML: https://mlsanity.com/paper/2409.11134/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2024.

## Built on by (most cited first, 1 of 1)

- [Bentkus-type asymptotic e-values](https://mlsanity.com/paper/2606.06332/) (2026)

## Abstract (arXiv)

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.