Bayesian interpretation of reliabilities

 

VanVleck (1979 and probably the Green Book) shows that (in our notation) $Var(u|\hat{u} )=(1-rel)Var(u)$. However,  $Var(u|\hat{u})$  is the same as $Var(u|y)$ . So we can put:

$1-rel=\frac{Var(u│y)}{Var(u)}$

And thus 

$rel=1- \frac{Var(u│y)}{Var(u)} = 1- \frac{PEV}{Var(u)}$

which is the usual equation. So, reliability is 1 minus the ratio of reduction from prior to posterior variance. It is an inbteresting Bayesian interpretation of which I was not aware.