p-Values, Bayes Factors, and Sufficiency

ABSTRACT

Various approaches can be used to construct a model from a null distribution and a test statistic. I prove that one such approach, originating with D. R. Cox, has the property that the p-value is never greater than the Generalized Likelihood Ratio (GLR). When combined with the general result that the GLR is never greater than any Bayes factor, we conclude that, under Cox’s model, the p-value is never greater than any Bayes factor. I also provide a generalization, illustrations for the canonical Normal model, and an alternative approach based on sufficiency. This result is relevant for the ongoing discussion about the evidential value of small p-values, and the movement among statisticians to “redefine statistical significance.”

KEYWORDS:

• Embedding model
• Exponential tilting
• Generalized Likelihood Ratio (GLR)

Acknowledgments

The author thanks Patrick Rubin-Delanchy and Christian Robert for their helpful comments on previous versions of this article; and a TAS reviewer, whose detailed comments on two versions of this article resulted in many improvements.

Funding

This research was supported by the EPSRC SuSTaIn Grant, reference EP/D063485/1.