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I was just in a lecture where my (bayesian evangelist) professor claimed that for questions like 'Is this a fair die?', frequency statistics gives an answer of {0, 1}, meaning that the probability is either zero or one. His justification was that if we imagine an infinite sequence of events stretching forward and count what percentage of them the die is fair in and what percentage of them the die is not fair in, we will get either zero or one.

This seems like a really weird interpretation of frequency statistics to me. Couldn't I count the number of dice I will ever hold and count how many are fair? What is I counted all the dice in the world and measured how many are fair? Wouldn't this make more sense?

Is he presenting a fair interpretation of frequency statistics? And, if so, why am I wrong?

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  • $\begingroup$ He asked about this die. Not other dice. You'd have to posit that the process of getting dice was equivalent to some kind of random sampling from a population of iid variates, but it's not. Different dice will have different probabilities (different biases). He's not interested in whether that transparent-gem Gamescience die is fair or whether that casino die is fair, he's interested in whether this particular Chessex die with the swirly pink and black on it is fair. He's proposing that a frequentist would assess the fairness of this die by tossing this die. $\endgroup$ – Glen_b Oct 20 at 23:57
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I would say that frequentist statistics doesn't give an answer to this question, at least not without some (at least partly) Bayesian addition or some additional definition of what "fair" means.

Frequentist statistics answers (among other things) "if this die was fair, how likely are these results?"

This was one of the reasons for developing Bayesian statistics in the first place.

Given a fair die, the most likely outcome is that each result will come up 1/6 of the time (assuming the number of rolls is divisible by 6) but, with even a reasonable number of rolls, the actual chance of this happening is small.

And, given any result and no other information, the best estimate is that the probabilities are what come out.

But what is a fair die? Probably no die has exactly 1/6 chance of each result; certainly the dice you can buy in a regular store don't have this -- they are just close. Casinos buy higher quality dice that are very close to fair, but exactness isn't possible.

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