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# 统计代写|统计推断代写STATISTICAL INFERENCE代写|STAT434 The Frequency Theory of Probability

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## 统计代写|统计推断代写STATISTICAL INFERENCE代写|The Principle of Indifference

Before commencing the historical development of the frequency theory, it will be useful to consider in what respects the principle of indifference, and along with it the classical definition of probability, offered itself as a target. Daston (1988) suggests that the principle had its origin in the concept of equitable contracts and that the symmetry of gaming devices seemed self-evidently to support the equiprobability assumption. In Hume, for whom mathematical, philosophical, and psychological aspects of probability were very close (Chap. 3), the mathematical or philosophical principle of indifference reflected primitive psychological states undisturbed by passions or other distorting influences. When probability replaced expectation as the fundamental concept, the principle began to look more circular; and when psychology came to be distrusted, by the French authors, as a basis for probability, the principle was undermined still further. Many applications, nevertheless, seemed straightforward, and the results, however technically difficult, noncontroversial.
The famous Rule of Succession was an exception. The rule was derived by a particular application of the principle of indifference, to probabilities themselves: Confronted with an unknown probability, in the absence of knowledge favoring one value over another, we can assume all possible values between 0 and 1 equally likely. The great power of the Law of Succession attracted critical scrutiny, which focused on the principle of indifference. Though it represented but one particular application, it had the effect, fairly or not, of discrediting the principle itself.

Its demise, as an accredited basis for probability assessment, was insured with further interesting paradoxes and dilemmas in the late nineteenth and early twentieth centuries. One example was already mentioned in Chap. 4: the ambiguity in whether different ratios of black to white tickets in the urn should be considered equally likely, or different constitutions of the urn. A better known example can be presented as follows. Suppose that the last stop Amtrak makes before it gets here is 120 miles away, and that the trip never takes less than 2 hours and never more than 4. (Suppose.) Without any knowledge of its track record, I might suppose all possible values for the duration of the trip between 2 and 4 hours to be equally likely. In particular, the trip should be as likely to take less than 3 hours as to take more than 3 hours. If I am ignorant of travel times, however, I am equally in the dark as to average speed, except that it will range between 30 and $60 \mathrm{mph}$. But a time of 3 hours corresponds to a speed of 40 , not $45 \mathrm{mph}$; so I cannot have a uniform distribution of probability for speed if I have one for time; yet my ignorance of the two variables is precisely equal.

## 统计代写|统计推断代写STATISTICAL INFERENCE代写|The Frequency Theorists

Keynes (1921/1973) accords actual priority for a frequency conception of probability to Leslie Ellis and to Cournot-to the former, for a paper read in 1842 (but not published until 1849), and to the latter for his Exposition de la Théorie des Chances et des Probabilités (1843); but John Venn (of diagram fame), whom they beat by 20-odd years, developed his theory more elaborately and so is usually given the roses. It is actually only in the most marginal sense that Ellis and Cournot may be identified as originators of the frequency theory. Certainly neither of them was setting out to purify the concept of probability of subjectivist connotations in the way that full-blown frequency theorists would. Cournot, in fact, was quite explicit about the reality and importance of epistemic probability and advocated a rather clearly dualistic view (see Chap. 3).

In the nineteenth century the frequency theory is associated most prominently, after Venn, with C. S. Peirce, and in the twentieth century with the positivist philosophers Richard von Mises and Hans Reichenbach. Karl Popper, who valiantly attempted to repair the defects of Mises’ theory, will be considered in this section, though he ultimately abandoned the frequency theory in favor of the propensity interpretation of probability.

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