Hello,
Just an explanation on something I may have explained badly. I see
we agree in the fundamental.
Correcting bias in that estimate should lead to
better sampling.
This is usually called "continuity correction"
http://en.wikipedia.org/wiki/Continuity_correction. The estimator
is not really biased, but because it is a quotient of integers it
requires a continuity correction specially when the integers are
small or zero is involved. That is included in the intervals I
suggested.
To use these results, you must make some assumption
about the underlying distribution of a move's probability
of winning.
That's the good news. You don't. There is no need to
understand what complex mechanism produces p. Only
that: same position == same p.
If you take a good look at your tests, they will make very specific null
hypothesis which in effect make at least some assumption about the
underlying distributions (or try to wash away all effects with the
central limit theorem).
Well, the "assumption" that p is estimated from the binomial because we
are counting Bernoulli experiments of constant p is a mathematically
sound method used universally. It does not require go knowledge, that's
what i meant. When n is big enough, the binomial converges to the normal
and that's what we use for inference.
Jacques.
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