Bet-Proofness as a property of Confidence Interval Construction Procedures (Not realized intervals)
There was a bunch of discussion over at Andrew Gelman's blog about "bet proof" interpretations of confidence intervals. The relevant paper is here.
The basic principal of bet-proofness was essentially that if a sample of data X comes from a RNG with known distribution that has some parameter , then even if you know exactly, so long as you don't know what the X values will be, you can't make money betting on whether the constructed CI will contain the (the paper writes this in terms of but the principal is the same since f is a deterministic function).
The part that confused me, was that this was then taken to be a property of the individual realized interval... "Because an interval came from a bet-proof procedure it is a bet-proof realized interval" in essence. But, this defines a new term "bet-proof realized interval" which is meaningless when it comes to actual betting. The definition of "bet-proof procedure" explicitly uses averaging over the possible outcomes of the data collection procedure but after you've collected and told everyone what it is, if someone knows and knows they can calculate exactly whether the confidence interval does or does not contain and so they win every bet they make.
So "bet-proof realized confidence interval" is really just a technical term meaning "a realized confidence interval that came from a bet proof procedure" however it doesn't have any content for prediction of bets about that realized interval. The Bayesian with perfect knowledge of and and the confidence construction procedure wins every bet! (there's nothing uncertain about these bets).