Author: Sune Fischer
Date: 15:38:06 11/24/03
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On November 24, 2003 at 12:59:39, Rémi Coulom wrote: >>I don't think that doesn't necessarily contradicts what you say though. > >No, it does not. Sorry I managed to fumble up that sentence, I think we agree though that the boolean question of whether p0>p1 does not depend on the number of draws. >>>If I cannot convince you, perhaps have a look at Rémi Coulom's paper, available >>>from http://remi.coulom.free.fr/ (inside >>>http://remi.coulom.free.fr/WhoIsBest.zip). One cite from that paper: >>> >>>"This proves that the likelihood that the first player is best does not depend >>>on the number of draws." >> >>Something to read tonight perhaps :) > >The paper is a bit mathematical, but the fact that the likelihood does not >depend on the number of draws can be explained intuitively rather easily: >imagine a game called "chess+" where no draw is possible: each time a game is >drawn, the two players start over from the initial position until one player >wins. Draws are not counted. For the exact same sequence of games, depending on >whether you consider they play chess or chess+, the score will be 1006-1000 or >6-0. Obviously, the likelihood that one is better than the other is the same. > >Of course, this is true only if the hypotheses are true: games are independent >random events, and the prior is uniform (which is reasonable in comp-comp >matches without learning). While reading it I managed to convince myself you are right. Intuitively it seems obvious that if the normalization constraint p0+p0.5+p1=1 is the only correlation between p0 and p1, then p0.5 isn't going to have any say in whether p0>p1 or not, right? The only exception is if p0.5=1, but then we wouldn't have a three parameter distribution in the first place which is sort of the assumption. -S. >I hope this message will save some mathematical reading for some. >Rémi
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