Saturday, December 6, 2014

Chernoff: "A career in statistics" (2014)

Chernoff makes some interesting remarks about the philosophy of statistics in his recent autobiographical essay.

First, an anecdote about the three classical decision criteria considered in decision theory:
I had always been interested in the philosophical issues in statistics, and Jimmie Savage claimed to have resolved one. Wald had proposed the minimax criterion for deciding how to select one among the many “admissible” strategies. Some students at Columbia had wondered why Wald was so tentative in proposing this criterion. The criterion made a good deal of sense in dealing with two-person zero-sum games, but the rationalization seemed weak for games against nature. In fact, a naive use of this criterion would suggest suicide if there was a possibility of a horrible death otherwise. Savage pointed out that in all the examples Wald used, his loss was not an absolute loss, but a regret for not doing the best possible under the actual state of nature. He proposed that minimax regret would resolve the problem. At first I bought his claim, but later discovered a simple example where minimax regret had a similar problem to that of minimax expected loss. For another example the criterion led to selecting the strategy A, but if B was forbidden, it led to C and not A. This was one of the characteristics forbidden in Arrow’s thesis.
Savage tried to defend his method, but soon gave in with the remark that perhaps we should examine the work of de Finetti on the Bayesian approach to inference. He later became a sort of high priest in the ensuing controversy between the Bayesians and the misnamed frequentists. (pp. 32–33)
He immediately moves on to one of his own more dismal conclusions about the issue:
I posed a list of properties that an objective scientist should require of a criterion for decision theory problems. There was no criterion satisfying that list in a problem with a finite number of states of nature, unless we canceled one of the requirements. In that case the only criterion was one of all states being equally likely. To me that meant that there could be no objective way of doing science. I held back publishing those results for a few years hoping that time would resolve the issue (Chernoff, 1954). (p. 33)
I haven't read the paper he is referring to here, but it seems like the text has jumbled up the conclusions: I think what he meant to say is that there is no single good decision function when we have infinitely many states, since the criteria essentially require us to use a uniform distribution. But I would have to check the details.

Finally, he moves on to a more on-record explication of his position:
In the controversy, I remained a frequentist. My main objection to Bayesian philosophy and practice was based on the choice of the prior probability. In principle, it should come from the initial belief. Does that come from birth? If we use instead a non-informative prior, the choice of one may carry hidden assumptions in complicated problems. Besides, the necessary calculation was very forbidding at that time. The fact that randomized strategies are not needed for Bayes procedures is disconcerting, considering the important role of random sampling. On the other hand, frequentist criteria lead to the contradiction of the reasonable criteria of rationality demanded by the derivation of Bayesian theory, and thus statisticians have to be very careful about the use of frequentist methods. 
In recent years, my reasoning has been that one does not understand a problem unless it can be stated in terms of a Bayesian decision problem. If one does not understand the problem, the attempts to solve it are like shooting in the dark. If one understands the problem, it is not necessary to attack it using Bayesian analysis. My thoughts on inference have not grown much since then in spite of my initial attraction to statistics that came from the philosophical impact of Neyman–Pearson and decision theory. (p. 33)

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