What is different about the q-value and local FDR when both are defined as posterior probabilities of the null being true?
For example in Storey (2010),
Under these modeling assumptions, it follows that q-value(pi) = mint≥pi Pr(Hi = 0|Pi ≤ t), which is a Bayesian analogue of the p-value – or rather a “Bayesian posterior Type I error rate.”
But the definition of the local FDR given just a few lines later seems quite the same
This connects the pFDR to the posterior error probability Pr(Hi = 0|Pi = pi), making this latter quantity sometimes interpreted as a local false discovery rate (Efron et al. 2001, Storey 2001).
I can see the mathematical definitions are different, but what's different conceptually?