Christian P. Robert and Joshua Bon’s contribution to the Discussion of ‘Safe testing’ by Grünwald, de Heide, and Koolen
讨论了e值与贝叶斯因子之间的联系，提出了一种在检验中定义最不利先验的形式化方法（定理1），并探讨了其在序贯蒙特卡洛和模型选择中的潜在应用。
Central to our interest in Bayesian foundations and reference priors, the connection made in this paper between e-values and Bayes factors, despite the cautionary added label of an ‘Almost Bayesian Case’, could bring 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. 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. Regarding the sequential directions of the paper, we wonder at its 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). In conclusion, we congratulate the authors on this worthwhile endeavour but remain agnostic as to whether or not the proposal may prove fruitful in the derivation of reference testing procedures. Funded by the European Union (ERC-2022-SYG-OCEAN-101071601) and by a Prairie chair from the Agence Nationale de la Recherche (ANR-19-P3IA-0001). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the Agence Nationale de la Recherche, the European Union, or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.