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BGonline.org Forums
A better approch? [LONG]
Posted By: MaX In Response To: A better approch? [LONG] (Maik Stiebler)
Date: Friday, 6 November 2009, at 3:31 p.m.
Maik wrote: I think MinCI does not give you anything close to Pbayes or PMC if you have two or more moves in tough competition for second best. As a simple extreme example (not that you are likely to ever see it in practice) imagine ten moves are practically tied for first place. If I understand correctly, MinCI would tell you that the primus inter pares has a 50% probability of being best?!
True, but in extreme cases, knowing that the confidence is 50% or 10% is not a big deal: they both mean that you have to extend the rollout. We are satisfied only with stuff in the 95% - 99.99% range, then I'm claiming that the 3 approaches (Pbayes, Pmc and minCI) are more or less equivalent.
In the minCI approach you could also make the (wrong) assumption that the events "play1 better than play2" and "play1 better than play3" are independent and multiply the CI of p1 vs p2 and p1 vs p3 in order to get a "lower bound" of the CI (not sure it really is a lower bound but it is in the approach).
I've simulated a 1000 trials rollout of 3 plays with equities .74 .72 and .70 (1001 bayes discretization points, 100000 montecarlo samples). Bayes gives 98.215%, mc 98.268%, CI the two bounds 98.283% - 98.470%. Anyone of the 3 is, to me, saying the same thing. Given all the assumptions behind each method, I wouldn't bother too much about 0.25% differences, especially since they are gonna squeeze if we extend beyond 1000 trials.
By the way, in my extremely naive python implementation, the Bayes approach (1001 discretization points) is much faster than the MonteCarlo approach. If we want to stop the RO on the prob of top play being better than all the others, then you must run the MC integration at each trial. On the other hand, you could just run the MC integration at the end of the RO, just to have the final probability, but this won't be a stopping criteria. But I'm not sure on the importance of the discretization in the bayes approach: it could well be that to match MC accuracy I need more than 1001 discretization points. Could be tough to evaluate.
I'm convincing myself that the best would be:
- use the bayes or montecarlo probability (whicever is faster) as criteria to stop the rollout
- At the end show:
* for the top play, Pmc (prob that the top play is better than ALL the others)
* for the other plays, the confidence interval of this play being better than the top one
This will give the two points of view: the global one (via the bayes/mc probability) and the play1 vs playN via the CI probability.
But an alternative could be to just use the CI approach to stop the rollout (similar to the current jsd stop criteria) and then compute the MonteCarlo estimate a posteriori (just once, at the end). if you're not satisfied with the resulting prob of the top player being best, just extend the rollout.
MaX.
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