A discussion of “A note on universal inference” by Tse and Davison

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Davidson and Tse carefully decompose the various sources of the power deficit experienced by universal inference (UI). We paid special attention to the case where there are many nuisance parameters, since we had hoped that the e-values from UI would be of use in popular high-dimensional models where it can be challenging to obtain valid p-values; for instance, it is now well established that classical asymptotic theory breaks down in high dimensional logistic regression. Unfortunately, the excellent article of Tse and Davidson shows that UI has very low power in problems with high-dimensional nuisance parameters. This short note explores ways to improve UI and make it more practical. In particular, Wasserman et al. (2020) built upon ideas in Grunwald et al. (2020) to introduce the reverse information projection (RIPR) split LRT, another e-value based on UI which is designed to increase power when testing composite null hypotheses. Unfortunately, the RIPR split LRT is often very challenging to compute. In this discussion, we suggest a simple modification called the quasi-RIPR split LRT. Although the quasi-RIPR split LRT is not always an e-value, it is in the multivariate Gaussian case (with a known covariance matrix) and is asymptotically equivalent to an oracle e-value for well-behaved parametric models. Most importantly, it is easy to compute and can dramatically improve the power of UI in settings with high-dimensional nuisance parameters.