Martin Larsson, Aaditya Ramdas, and Johannes Ruf’s contribution to the Discussion of ‘Safe testing’ by Grünwald, de Heide, and Koolen

原文摘要

We congratulate the authors on a stimulating article that has inspired many follow-up works. We discuss the central concept of growth-rate optimality (GRO). For a point null and alternative, the GRO e-variable is their likelihood ratio (Shafer, 2021). The generalization to composite nulls takes centre stage in Grünwald et al. (2024). Their use of the reverse information projection (RIPr) builds on Li (1999), and has been recently extended in Lardy et al. (2023). Inspired by their centrality, we report on our efforts to significantly generalize these results. Our preprint (Larsson et al., 2024) considers testing a composite null P against a point alternative Q⁠. We establish a powerful result: under no conditions whatsoever (e.g. reference/dominating measure), there always exists a unique ‘optimal’ e-variable that we call the numeraire X*⁠: for every other e-variable X, we have EQ[X/X*]≤1⁠. In particular, X* also satisfies EQ[log(X/X*)]≤0 (log-optimality), as well as EQ[1/X*]≤1⁠. X* also identifies a particular measure P* by the definition dP*/dQ=1/X*⁠. In general, P* is a subprobability measure in the ‘bipolar’ of P⁠. In particular, X*=dQ/dP* is a generalized likelihood ratio of Q against P⁠, and is the only e-variable with such a representation. Further, P* coincides with the RIPr when additional assumptions are made that are required for the latter to exist [such as Q≪P for some P∈P⁠, σ-convexity of P⁠, and finite maximum description gain in Lardy et al. (2023)]. Thus, we contend that going forward, absent any assumptions on Q,P⁠, the above should be the default definition of the RIPr. We also present a more general optimality theory that studies general concave utilities (beyond the logarithm), exemplified by a reverse Rényi projection, which also exists under no conditions. Surprisingly, our results imply that the universal inference e-variable (Wasserman et al., 2020) is (essentially) always inadmissible. Practically, the numeraire is now on a somewhat equal footing with universal inference. While the latter seems easy to apply, it is has only been applicable in a few nonparametric situations thus far (like testing log-concavity). In contrast, we explicitly calculate the numeraire and RIPr in many nonparametric settings where there is no reference measure, and universal inference does not apply; we explore several such cases in the paper (testing symmetry, sub-Gaussian means, bounded means). We are currently working to extend these ideas to composite alternatives using standard techniques like the method of mixtures, and to sequential settings using the mixture and plug-in methods; see Ramdas et al. (2023) for details.

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