Joshua Bon and Christian P. Robert’s contribution to the Discussion of ‘Safe testing’ by Grünwald, de Heide, and Koolen

原文摘要

The central proposal found in this quite original paper could prove relevant for the so-called objective Bayes approach (Berger, 1985; Robert, 2001) as it may open a way to deriving prior distributions from (frequentist) first principles. While some Bayesian schools refuse to consider the concept of testing the veracity of a null hypothesis, most do propose approaches and methodologies to address this challenge. These mainly rely on Jeffreys’ formalization via two prior distributions (Jeffreys, 1939; Wrinch & Jeffreys, 1919) via what would later come to be the Bayes Factor (Ly et al., 2016; Robert et al., 2009; Wagenmakers & Ly, 2022). In his Theory of Probability, Jeffreys constructs the equivalent of his point estimation prior for tests, although it had little impact on the field until rediscovered by Bayarri and Garcia-Donato (2007). We commend the points put forward to support e-values, behaviour under optional continuation [by a martingale reasoning] interpretation as ‘evidence against the null’ as gambling in all cases preserving frequentist Type I error guarantees e-variables turn out to be Bayes factors based on the right Haar prior (rather than at times on highly unusual (e.g. degenerate) priors?) e-variables need more extreme data than p-values in order to reject the null since they are mostly worthwhile, even though (b) is vague and (c) is frequentist. The link of the theory of e-values with right Haar priors reminds us of Pitman’s derivation of the best equivariant estimator (Robert, 2001), even though the fact that Haar priors are improper for the most part proves a major hindrance in a testing situation (DeGroot, 1973). The optional continuation in (a) is an even stronger argument—if from a Bayesian viewpoint—since it is been used to defend the Bayesian paradigm. On the other hand, if sequentially testing e-values, only the first e-value will be interpretable as a coherent Bayes factor. That is, in sequential testing, the product of the e-values does not share the same properties as a product of Bayes factors (i.e. cancellation of terms as a telescoping product). Central to our interest, point (d) above, despite the cautionary added label of an ‘Almost Bayesian Case’, brings a formal way to define least favourable priors (Berger, 1985) in a testing environment, albeit in no clear connection with the reference priors of Bayarri and Garcia-Donato (2007). This derivation, formalized as Theorem 1, provides a reversal of the typical prior choice in testing. A more common stance is to hold the (subjectively chosen) prior distribution on H0 as a starting point—this is an obviously known entity. One may wonder if a dual perspective, i.e. when starting from the prior distribution on H0⁠, would prove fruitful, e.g. in leading to an optimal prior on H1⁠. The paper seems to conclude at a perfect duality with the minimax–maximin result. This also relates to the projection priors we defined in Goutis and Robert (1998) and Dupuis and Robert (2003). As a plus, however, the approach found therein allows for improper priors on the nuisance parameters and thus somehow brings a stronger justification than Berger et al. (1998) in favour of the permanence of an identical measure (on the nuisance parameters) across hypotheses and models. In conclusion, we congratulate the authors on this endeavour but it remains unclear to us (as Bayesians) (i) how to construct the least favourable prior on H0 on a general basis, especially from a computational viewpoint, and, more importantly and (ii) whether it is at all of inferential interest (i.e. whether it degenerates into a point mass). With respect to the sequential directions of the article, we also wonder at the potential connections with sequential Monte Carlo, for instance, towards conducting sequential model choice by constructing efficiently an amalgamated evidence value when the product of Bayes factors is not a Bayes factor (see e.g. Buchholz et al., 2023). Funded by a Prairie chair from the Agence Nationale de la Recherche (ANR-19-P3IA-0001) and by the European Union (ERC-2022-SYG-OCEAN-101071601). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the a Prairie chair from the Agence Nationale de la Recherche (ANR-19-P3IA-0001), the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）