There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"

TL;DR

This paper discusses Shafer's betting perspective on statistical testing, emphasizing the need for real-world examples and clarification of the betting score's interpretation in relation to study goals and assumptions.

Contribution

It provides a detailed rationale for using the surprisal (logworth) as a betting score, clarifying its justification in the context of information statistics.

Findings

Surprisal (logworth) is a well-motivated betting score.

Justification of betting scores depends on study goals.

Uncertainty in sampling models affects score interpretation.

Abstract

Shafer (2021) offers a betting perspective on statistical testing which may be useful for foundational debates, given that disputes over such testing continue to be intense. To be helpful for researchers, however, this perspective will need more elaboration using real examples in which (a) the betting score has a justification and interpretation in terms of study goals that distinguishes it from the uncountable mathematical possibilities, and (b) the assumptions in the sampling model are uncertain. On justification, Shafer says 'No one has made a convincing case for any particular choice' of a score derived from a P-value and then states that 'the choice is fundamentally arbitrary'. Yet some (but not most) scores can be motivated by study goals (e.g., information measurement; decision making). The one I have seen repeatedly in information statistics and data mining is the surprisal, logworth or S-value s = -log(p), where the log base determines the scale. The present comment explains the rationale for this choice.