Math CentralQuandaries & Queries


Question from Paul:

Suppose I have a dice in my hand that I am about to roll. The probability that I roll a six is, all things being equal, 1/6. I accept that.

Suppose now the I roll the dice and immediately cup my hand over the result. What is the probability that I have rolled a six? People seem to want to say it is still 1/6. But it can't be can it!? It is surely either 1 or 0, depending on whether I have in fact rolled a six?


This is a long-standing question in the foundations of probability theory, and probabilists are not in agreement over the answer!

A "frequentist" statistician would agree that once the die is rolled the concept of probability does not apply any more. A "Bayesian" would say that probability can describe an individual's partial knowledge of a state of events, including but not limited to the outcomes of randomized experiments that have not yet taken place.

Criticism of the Bayesian viewpoint tends to center on the subjective nature of what it measures. There is also a pragmatic element, in that Bayesian statistical practice seems to require somewhat more advanced math (about two years of calculus) than frequentist practice, and the university departments whose students account for most elementary stats learning (psychology, sociology, commerce) would lose most of their students to other universities if they required them to take so much math. The marketplace demands low-math statistics, and frequentist stats currently fits the bill better. (This could change with future software, I suppose.)

Criticism of the frequentist viewpoint points out that its workhorse procedure, hypothesis testing, answers a question ("how improbable is it that data differing at least this much from the values predicted by the null hypothesis would be observed if the null hypothesis were really true?") that is not the one ("how probable is the alternative hypothesis given the data?") that people really want to answer. It has also been pointed out that if you [without looking] offer a frequentist believer a bet at better than 5:1 odds that your static covered die is a 6, [s]he will probably take you up on it, whereas if the odds are worse than 5:1 [s]he won't; in other words, even frequentists seem to use Bayesian probability in decision making even if they refuse to give it the name.

Finally, it is not clear, at least to me, what is meant by the crucial frequentist concept of a "random experiment". In a deterministic, Newtonian model of physics, the fall of the die would have been determined from when you threw it - and maybe even from when you decided to do so or before. You yourself perhaps wouldn't have been able to predict it, but in the frequentist view "ignorance isn't probability." One could invoke quantum mechanics and perhaps also free will here, but it is not clear why or how either of these affects the validity of statistical theories.

A very readable (if occasionally slightly aggressive) book arguing for the Bayesian viewpoint is Edwin T. Jaynes' "Probability Theory: The Logic of Science".

Good Hunting!

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