No, the point is when you do random move picking the probability you assigned to the moves can only be respected only at _inifinite_ number of games. Until then the estimate will hinge on the fact that how representative a move is picked up in my case (since I use constant multiplier), but in your case it requires perfect estimation of multipliers so it can give bad results.Not true. The method I and Michel propose will not give "bad means". Their mean is exactly equal to the true size of the tree, no matter how wrong the assumptions were that went into deriving the p_i. Wrong assumptions just mean that you converge to that correct mean more slowly than with correct assumptions. But you cannot avoid making assumptions. Making 'no assumptions' on the relative sub-tree sizes is equivalent to the assumption that they are on averageof equal size(for which the optimum would be homogeneous sampling.) That assumption can be just as wrong as any other. And inparticular with LMR'ed trees it is a very inferior assumption.
This is clear with playing one game, I have to pick the average move, and you guys have to get each proportion right. My heuristic is much easier to do than yours because it doesn't ask for too much. If we play only 100 games on 5 moves we can not assume each move to get 20. That only happens on a infinite number of games. For low number of games you must have the correct multiplier all the time. This is what I am objecting too. You can't make such a good estimate at lower depths without a very very good heuristic... A simple example is when you made a terrible mistake in proportioning, say the moves were 100 25 25 100 but you use weights like which assumes an LMR kind of node distribution, clearly it will give bad results. Even when your relative proportions are wrong slightly will still give bad results. So when you stop the simulations abruptly, your estimate will can give very bad results. But if I pick average (representative) values more often, I get good estimates even at very low number of games.