Proposer of the vote of thanks to Waudby-Smith and Ramdas and contribution to the Discussion of “Estimating the means of bounded random variables by betting”

The authors derive nonasymptotic anytime-valid confidence sequences for the mean of a se- quence X1, X2, . . . of bounded random variables. When put to practice, their new methods beat the best known bounds, sometimes by vast margin — even for the fixed-sample size, not anytime- valid setting. It is rare in statistics that one can get such substantial improvements on a decades-old problem, and I congratulate Waudby-Smith and Ramdas on this remarkable achievement. It il- lustrates once more the relevance of e-process-based anytime-valid methods [Gr¨unwald et al., 2024] even when anytime-validity is not required. In particular their results have repercussions for PAC- Bayesian machine learning theory, which relies on concentration bounds for bounded i.i.d. Xt — the main (but not only) setting the authors (WSR from now on) consider and on which I will also focus. So, let X1, X2, . . . be i.i.d. ∼ P with P an arbitrary distribution on [0, 1]. The null, denoted Pμ, consists of all distributions on [0, 1] with some fixed mean μ. We want to test whether the mean is μ, against alternative S μ′̸ =μ Pμ′.

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