Extended Likelihood
Yudi Pawitan, Youngjo Lee · 2024
We describe extended likelihood and hierarchical likelihood as the unifying tools for dealing with complex models that involve fixed and random parameters. We explain how extended likelihood is different from classical likelihood. We show the difficulties that can arise when using extended likelihood, primarily due to the lack of invariance with respect to the transformation of the random parameters. The hierarchical likelihood is introduced as a special extended likelihood that avoids these problems and gives optimal estimators and predictors for both fixed and random parameters. Confidence is shown to be an extended likelihood; this connection expands confidence&s;s utility beyond the confidence-interval contexts. In various scenarios, confidence captures the sense of uncertainty about realized parameters or events. The wallet game paradox demonstrates how the use of probability language can lead to paradoxes; however, extended likelihood avoids such issues.