[BillChen]

It seems from a lot of the literature, including the Yale stats page and the Association of Physicians that the definition of confidence interval as the interval with the specified value of the parameter in that range. I also asked a couple of my mathematician friends who are doing research outside of statistics who gave a similar definition.

But I think we agree on the point Jerrod was trying to make that a two std deviation radius window from the mean does not mean a 95% probability that the win rate falls within this window, and I don't think this point is well communicated to the world. Without making a value judgement on whether the definition is misleading, I would say most poker players who know about confidence intervals are misled.

Also maximum likelyhood estimators are not always unbiased. A MLE represents the mode or peak of the parameter distribution which may be to the right or to the left of the mean.

I agree it's nice to know what your distribution of observed win rates is given a true win rate--as in most things trying to go backwards from an observed win rate and other information to make a guess at a true win rate is much harder and less well defined, but it's much more important to the poker player. This reminds me of the story of two guys lost in a balloon who ask where they are. "You're in a hot air balloon." "He must have been a mathematician. What he said is precisely correct, but is of no use to us."

Bill Chen