Just to add an additional voice here, I would view that as incorrect in this context, instead referring to the thing that the CEE is saying. The way I’d try to clarify this would be to put the variables varying in the expectation in subscripts after the E, so the CEE equation would look like ED[P(H=hi|D)]=P(H=hi), and the PPI inequality would be E(H,D)[P(H|D)]≥EH[P(H)].
Just to add an additional voice here, I would view that as incorrect in this context, instead referring to the thing that the CEE is saying. The way I’d try to clarify this would be to put the variables varying in the expectation in subscripts after the E, so the CEE equation would look like ED[P(H=hi|D)]=P(H=hi), and the PPI inequality would be E(H,D)[P(H|D)]≥EH[P(H)].