An axiomatic investigation of the e-value as a measure of support for hypotheses - Cavaliere, Yasmin Ferreira (2024)
Testing a hypothesis is fundamental in any scientific investigation or data-driven decision-making process. Since Pearson systematized the use of hypothesis testing, this statistical procedure has significantly contributed to the development of various theories of statistical inference. Common approaches to hypothesis testing include significance tests, uniformly most powerful (UMP) tests, generalized likelihood ratio (GLR) tests, and Bayesian tests. However, practitioners often adopt evidence measures, such as p-values, Bayes factors, and likelihood ratio test statistics, as they provide a more nuanced understanding of hypotheses beyond the reject-non reject approach by Neyman and Pearson. This study proposes an axiomatic development of belief relations representing the extent to which sample data support a hypothesis, consistent with the Onus Probandi Principle. We specifically examine whether Pereira-Stern evidence (e-value) is a reasonable representation of how much a sample supports a hypothesis in discrete parameter and sample spaces. We show that the discrete evidence from Pereira-Stern is a measure of support of hypothesis that is a representation of a belief relation, being compatible with various probability distributions which are the opinions/beliefs of many Bayesian decision-makers agents. Future research will address open issues, including less restrictive conditions for enumerable cases, generalizing axiomatization to non-enumerable spaces, and associating hypothesis testing with belief relations.