Abstract
1. In the so-called inductive (or mathematical) statistics rules of inference of various kinds are formulated. The fundamental class of such rules is known under the technical name of rules of parametric inference with one parameter unknown, experimentation fixed, no a priori distribution assumed. They can be described as follows.
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“This much is clear: the utilities should reflect the value or disvalue which the different outcomes have from the point of view of pure scientific research rather than the practical advantages or disadvantages that might result from the application of an accepted hypothesis, according as the latter is true or false. Let me refer to the kind of utilities thus vaguely characterized as purely scientific, or epistemic, utilities” (Carl G. Hempel, ‘Inductive Inconsistencies’, Synthese, Vol. XII, 1960, No 4, p. 465.
It should be remarked that in the definition below we use a shorthand notation again: as the outcome of the experiment is an n-dimensional random variable, the sign of addition stands for multiple addition (or integration) with respect to n variables.
Cf. for instance, K. Szaniawski, ‘Some Remarks Concerning the Criterion of Rational Decision Making’, Studia Logica, IX, 1960. (see this volume, pp. 114-127.]
It would not, in general, be possible to justify a rule by showing that its efficiency is above a certain preassigned level, because of the fact that this level can always be raised (or lowered) by suitably redefining r.
K. Ajdukiewicz, ‘Zdania pytajne’ [Interrogative Sentences] in Język i poznanie [Language and Knowledge], PWN, Warszawa 1960. Reprinted from Logiczne podstawy nauczania, 1938. See also D. Harrah, ‘A Logic of Questions and Answers’, Philosophy of Science, Vol. 28, 1961, No 1.
In such a case the end has obviously the ‘all or nothing’ character, and we can assign any two numbers to its degrees of realization. It is most convenient to choose 0 and 1 for this purpose. The efficiency of a rule is then a number in the interval (0,1).
Cf. K. Szaniawski, ‘On Some Basic Patterns of Statistical Inference’, Studia Logica, XI, 1961. [see this volume, pp. 70-79.]
The conclusion obtained by means of such a rule is called in statistical literature a confidence interval at 1 — a level. The method has been introduced by J. Neyman.
Thus, e.g., the maximum most efficient d would be the one with e(d ω0) = 1/2, and the Laplace most efficient d would lead to the rejection of ω0, whatever the outcome of the experiment.
See A. Wald, Sequential Analysis, J. Wiley, New York 1947, p. 27–29.
Ibidem.
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Szaniawski, K. (1998). A Pragmatic Justification of Rules of Statistical Inference. In: Chmielewski, A., Woleński, J. (eds) On Science, Inference, Information and Decision-Making. Synthese Library, vol 271. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5260-0_11
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