Seconder of the vote of thanks to Waudby-Smith and Ramdas and contribution to the Discussion of ‘Estimating means of bounded random variables by betting’

原文摘要

The paper proposes an elegant new framework for deriving time-uniform confidence intervals for the mean of a sequence of random variables. The proposed technique is based on a game-theoretic view of statistical inference, rephrasing the problem of building confidence intervals as identifying a set of coin-betting games with a certain ‘plausibility’ property: if one can get ‘implausibly rich’ in a game by betting on the mean being lower (resp. higher) than a certain threshold, then the mean must be higher (resp. lower) than said threshold. A confidence interval can then be derived as the interval between the lowest and highest values that cannot be ruled out as ‘implausible’. This game-theoretic view extends the pioneering work of Ville (1939) and Shafer and Vovk (2019) who used similar betting games to lay an alternative set of foundations of probability theory that is free from measure theory. The authors of this work develop a set of new concentration inequalities from this framework that are both aesthetically pleasing and empirically tight. In many ways, the framework appears to be so natural and profound that the reader wonders why this connection with betting has not been exploited in this depth previously. Indeed, the language of betting is so closely tied with the theory of test martingales that one wonders why almost all past work on deriving confidence intervals (and more generally, testing) have focused on using supermartingales that inevitably lead to at least some looseness when used for this purpose. In other words, I believe that the authors have contributed a valuable fundamental insight which will undoubtedly lead to many interesting future results. Besides this appraisal, I would also like to mention some limitations of the results presented in the paper and particularly highlight connections with the concurrent work of Orabona and Jun (2021) that proposes a similar framework for deriving confidence sequences. One slightly unsatisfying aspect of the paper is that, while the authors develop a new theory of mean estimation through the lens of playing a sequential game, they do not seem to make full advantage of this connection by going ‘all in’ on this connection. In particular, most of the proposed estimators are based on betting heuristics rather than betting strategies with already existing theoretical guarantees on their growth rate (or other properties of interest). There are several such algorithms for a range of known betting games, and I am surprised to see that these are only mentioned as an afterthought in the appendix (if at all). In particular, it remains unclear what betting algorithm one should use to, say, minimise the confidence width. The authors criticise the use of principled betting strategies, saying that those are all optimised for worst-case price relatives, and the i.i.d. case considered in the paper should be much easier to handle. On one hand, this is hardly an appropriate justification to replace principled methods with heuristics. On the other hand, this argument is not entirely well-founded, as several well-studied betting strategies are ‘equalisers’ in the sense that they have the same regret for all sequences (i.e. worst-case or i.i.d.). See the detailed discussion on this issue by Orabona and Jun (2021). The confidence intervals depend on the order in which the data are presented to the algorithm. This is natural for non-i.i.d. data but is hard to justify when the data are i.i.d. The authors propose a ‘derandomisation’ strategy consisting of shuffling the data a number of times, executing the algorithm on all resulting sequences, and aggregating the results. Besides being somewhat unsatisfactory from a conceptual point of view, this approach also destroys the efficiency that was part of the magic of the proposed method. I wish to point out that this aspect has been addressed very neatly by Orabona and Jun (2021), whose reduction to the universal portfolio optimisation method of Cover and Ordentlich (1996) produces estimates that are independent of the order of the data without such derandomisation. The method is limited to random variables bounded almost surely in [0,1]⁠. This seems like an inherent limitation, although it seems likely that it can be removed either by combining the current approach with the ‘moment truncation’ estimator of Catoni (2012) (as one of the authors did in a related paper, Wang & Ramdas, 2023). Still, it is unclear if such extensions can retain the appealing ‘purity’ of the approach proposed in the present paper, or if a more elegant and natural treatment of unbounded random variables would be possible. I wish to emphasise that these latter comments are not meant to diminish the value of the contributions of the paper but rather to highlight some outstanding issues that future work has to address. There are of course several more open questions left behind that this note can hardly do justice to. That said, opening so many potential directions for future research is a sign of great fundamental work, and I wish to congratulate the authors (as well as the concurrent pioneers; Orabona & Jun, 2021) for bringing this set of tools to the forefront of statistical inference. The vote of thanks was passed by acclamation.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）