Hi
regarding what you said about confidence intervals, how after only a small number of actions are observed that these can extend to the negative side of the number spectrum etc etc
I think you are confused. "Check-raise" is an action, one of several potential actions a player may take, and of course also relies on conditional probabilities (i.e., one may be more likely to check raise in positions where they may have strong holdings, and they may have strong holdings more often when they raise preflop, for example).
That aside, being that check-raising is one of a number of potential options and that we are concerned with the likelihood of it occurring, the statement that "the confidence interval extends into negative numbers" is absolutely ridiculous, and is indicative of nothing. Stop trying to interpret such things in a linear manner. We are dealing with probabilities; think of a normal or [ exp(.) ] / [ 1 + exp(.) ] i.e. logistic distribution instead. These will always fit probabilities between 0 and 1.
In conclusion, you are misspecifying the problem at hand.
Also I should mention I skimmed most of your post etc etc, no offense. and thanks for contributing anyway.
EDIT: Oh yeah I do agree with the basic premise that larger sample sizes are needed etc. But I don't have much problem with people making inferences based off small-sample properties either, like when sitting with a group of opponents for less than an hour and having to decide whether a given player 3-bets too often or whatever.
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