Analytical Methods in Probability Theory. Proc. conf. by Daniel Dugue, E. Lukacs, V. K. Rohatgi

By Daniel Dugue, E. Lukacs, V. K. Rohatgi

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Extra resources for Analytical Methods in Probability Theory. Proc. conf. Oberwolfach, 1980

Sample text

1148). Therefore, when the answer is either “yes” or “no” the question is settled. 5 In this case, the question will be to try to translate verbal expressions like “very much probable” or “much probable” or “little probable” or “very little probable,” and so forth, into numerical terms. All these concepts are very general and despite my efforts to make them concrete, they might not yet be entirely clear to you. And, since we still have five minutes at our disposal, I would like to use this time to answer your questions.

Pn , E 1 , . . , E n )) only if p1 = q1 , . . , pn = qn . ’ The history of proper scoring rules begins in 1950 with an article by the meteorologist Glenn Wilson Brier (1950), who introduced the so-called Brier’s rule to be applied to meteorological forecasts. It is obtained by putting: f (q1 , . . , qn , E 1 , . . ” Carnap’s ideas have been later fruitfully developed by Roberto Festa (1993). Carnap’s rule turns out not to be proper in the sense defined above (Br¨ocker and Smith, 2006), but it would be proper if the class of admissible probability evaluations were so restricted that only those evaluations that satisfy some symmetry constraints (actually required by Carnap) were allowed.

And proper scoring rules are the only adequate instrument by means of which that degree of belief can be measured. The procedure to obtain such a measure is as follows. A person X is asked to indicate her own probability evaluations. Such a person is warned that she will receive a score depending on the numbers she has stated. Scores, on the other hand, are devised ad hoc so that it is advantageous for X to indicate exactly those numbers which correspond to her own degrees of belief, as this would minimize the prevision of the penalization.

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