Is the Tail Area Useful as an Approximate Bayes Factor?

Abstract

Inequalities are given relating the tail area for a nested sharp hypothesis to the Bayes factor based on the event of “significance” considered as data. This Bayes factor based on an insufficient statistic is, in turn, expressed as a weighted average of full-data Bayes factors. Lindley's “statistical paradox” is generalized and other comparisons made in the normal sampling context. A new Bayesian interpretation is given for the traditional two-tailed critical level. An example and the discussion suggest a negative answer to the question in the title.

Key Words:

• Tests of hypotheses
• Significance tests
• Odds ratios
• Bayes factors
• Statistical inference
• Bayesian inference