Shackle on case probaility
It is interesting how Shackle and Mises are partners in this matter. The authors of G. L. S. Shackle discuss Shackle's rejection of the notion that a singular event can meaningfully be said to have probability X.
Shackle puts forward an example where England and Australia are to have a cricket match. But instead of the usual coin toss deciding who bats first, England has managed to get the matter on the toss of a die, where a one will mean Australia bats first, and any other number, England will do so. He asks, "Can we now give any meaningful answer whatever to the question 'Who will bat first?' except 'We do not know'?" (p. 63)
The authors reject Shackle's agnosticism here, claiming: "Of course, the right answer is 'England will, most probably'" (p. 64).
But this simply begs the question that Shackle was raising: In the case of a single event, what exactly do we mean when we assert its probability is X? Of course everyone agrees that if a long series of matches takes place with this method of determining who bats first, England will in the majority of cases. In defense of their view, the authors note that a bettor will want to place his bet on England. But that truth appears to rely on the fact that, over the long run, a repeated series of such bets will pay off, i.e., it seems to rely on the frequency-based interpretation of probability, an interpretation that Shackle endorses!
I don't assert that Shackle was right here, but only that his argument against case probability cannot be dismissed with a mere "Of course!"
Shackle puts forward an example where England and Australia are to have a cricket match. But instead of the usual coin toss deciding who bats first, England has managed to get the matter on the toss of a die, where a one will mean Australia bats first, and any other number, England will do so. He asks, "Can we now give any meaningful answer whatever to the question 'Who will bat first?' except 'We do not know'?" (p. 63)
The authors reject Shackle's agnosticism here, claiming: "Of course, the right answer is 'England will, most probably'" (p. 64).
But this simply begs the question that Shackle was raising: In the case of a single event, what exactly do we mean when we assert its probability is X? Of course everyone agrees that if a long series of matches takes place with this method of determining who bats first, England will in the majority of cases. In defense of their view, the authors note that a bettor will want to place his bet on England. But that truth appears to rely on the fact that, over the long run, a repeated series of such bets will pay off, i.e., it seems to rely on the frequency-based interpretation of probability, an interpretation that Shackle endorses!
I don't assert that Shackle was right here, but only that his argument against case probability cannot be dismissed with a mere "Of course!"
Gah Gene! Probability when you get down to it is a quagmire!
ReplyDeleteThis is a case where the Bayesians seem right. Probability statements are about degrees of belief. It sure looks that way here. But it is also a case where Bayesians have no answer the point you raise. The degree of belief is based on the long term frequency.
This is why, for all its appeal, I am not (yet) a Bayesian.
But i will take a bet at better than 1 to 5 on England batting first.