https://orcid.org/0000-0001-7296-6275
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Andrej Srakar, Andrej Srakar’s contribution to the Discussion of ‘Safe testing’ by Grünwald, de Heide, and Koolen, Journal of the Royal Statistical Society Series B: Statistical Methodology, Volume 86, Issue 5, November 2024, Page 1159, https://doi.org/10.1093/jrsssb/qkae062
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Paper presented by Grünwald, de Heide, and Koolen presents an extension of e-values to the testing of composite hypotheses in a context they label ‘safe testing’. It discusses what should be a merging of the Fisherian, Neymanian, and Jeffreys-Bayesian interpretations to the testing of hypotheses. Proposed tests based on e-values are safe, i.e. they preserve type-I error guarantees, under proposed optional continuation where the decision to perform a new study may depend on previous outcomes.
I provide short remarks related to probability aspects and possibilities in the proposed approach. Namely, construction of the approach is based on supermartingales and relation to previously studied e-processes (Ramdas et al., 2021). Brownian motion is mentioned at the end of the article appendix but not contextualized in more detail. What I question (this is also based on the illustrative Figure 1) is about possible additional extensions in a probability context. For example, when one would want to address the context of composite hypotheses in their dependencies, approaches and stochastic models based on interacting particle systems (such as simple exclusion or simple inclusion processes, or many extensions therein) could be interesting and useful to include. Literature in this area has previously studied relationship to martingales (e.g. Delgadino et al., 2021; Oelschläger, 1984). Other probability theory contexts could also be addressed for future research on e-values.
