You say this can not be but what happens is the following. If I compute the weights from the same sequence and do that only at the root it is still unbiased. And if do it in 2-ply , 3-ply searches it becomes "visibly" more and more biased. This is a fact I can prove by running experiments. Peter also observed something similar.I think this cannot be. Unless of course you reuse the data 200 125 100 50 25.
After having generated the proportions from it you should discard this data
(but you can keep an estimate generated from it, please read my post).
So do we agree the problem is there if summation is done elsewhere other than the root? At least that is what I observed, maybe the bias is small to see when summation is done at the root? Are you saying the bias is still there if I reuse the data even at the root?
EDIT:
On second thought I think you were being very unfair to me when you discard the points I am trying to make. Well you also had confusions with my proportions and weights right? I was _honestly_ (can't stress enough) trying to understand what is happening, but if we are mis-communicating well it is better to stop it. Anyway both methods are practically unbiased so there is no issue. Just that I don't understand (you seem to understand it) doesn't mean anything. Lets move on with other tests, no hard feelings.