de Finetti assumes conditioning. If I am taking conditional expectations, then iterated expectations (with different conditionings) is very useful.
But iterated expectations, all with the same conditioning, is superfluous. That’s why I took care not to put any conditioning into my expectations.
Or we can criticize the probability-of-a-probability musings another way as having undefined filtrations for each of the stated probabilities.
de Finetti assumes conditioning. If I am taking conditional expectations, then iterated expectations (with different conditionings) is very useful.
But iterated expectations, all with the same conditioning, is superfluous. That’s why I took care not to put any conditioning into my expectations.
Or we can criticize the probability-of-a-probability musings another way as having undefined filtrations for each of the stated probabilities.