Parameter Factorization and Inference Based on Significance, Likelihood, and Objective Posterior

Abstract

The concepts of significance, likelihood, and objective posterior have wide ranges of application in statistics. For certain very simple applications--single location variable--there has been recognition that the three concepts produce equivalent numerical results, specifically the equality of observed level of significance, integrated likelihood extreme values, and integrated objective posterior extreme values. The most general model permitting the use of the three concepts for the full parameter, and indeed for component parameters, is a structural model (or probability-space model). This paper examines the three concepts for a structural model and shows for both the full parameter and for component parameters the essential equivalence of observed level of significance, integrated likelihood extreme values, and integrated objective posterior extreme values.