On Sep 24, 2008, at 2:40 PM, Jacques Basaldúa <[EMAIL PROTECTED]>
wrote:
> "The approach of this paper is to treat all win rate estimations
as independent estimators with
additive white Gaussian noise. "
Have you tried if that works? (As Łukasz Lew wrote "experimental set
up would be useful") I guess
there may be a flaw in your idea, but I am not a specialist. I will
try to explain it.
I will try to address your concerns in the next revision of the paper.
I'll discuss computing the properties of the estimators and add extra
rigor using Bayes Theorem (as Łucasz suggested). More comments below.
If it wasn't for the fact that the tree is learning, the probability
of a playout through a node to win
would be constant each time the node is visited. This is, of course,
a simplification because the tree
does learn, but, at least between playouts that are not very distant
in time, it is true. So my argument
holds to some (I guess, much) extent. The same applies to the RAVE
estimator which is also the result
of counting wins (assume P(win|that move) = constant) and dividing
by some appropriate sample size.
Therefore, these estimators follow a binomial distribution. It does
converge to the normal, but with
some fundamental caveat: Unlike the normal in which mean an variance
are independent, in this case
the variance is a function of p.
That's all true, and does conflict with my simplistic RAVE bias
discussion...
The variance of the binomial = n·p·(1-p) is a _function of p_.
That's when binary samples are summed. When averaging, it's p•(1-p)/
n. My RAVE discussion essentially used p=1/2.
Therefore, the variance of the normal that best approximates the
distribution of both RAVE and
wins/(wins + losses) is the same n·p·(1-p)
See above, it's slightly different.
If this is true, the variance you are measuring from the samples
does not contain any information
about the precision of the estimators. If someone understands this
better, please explain it to
the list.
This will get covered in my next revision. A proper discussion is too
much to type with my thumb..._______________________________________________
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