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The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical tendency of something to occur or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories〔 The taxonomy of probability interpretations given here is similar to that of the longer and more complete Interpretations of Probability article in the online Stanford Encyclopedia of Philosophy. References to that article include a parenthetic section number where appropriate. A partial outline of that article: * Section 2: Criteria of adequacy for the interpretations of probability * Section 3: * * 3.1 Classical Probability * * 3.2 Logical Probability * * 3.3 Subjective Probability * * 3.4 Frequency Interpretations * * 3.5 Propensity Interpretations〕〔 "There are several schools of thought regarding the interpretation of probabilities, none of them without flaws, internal contradictions, or paradoxes." (p 1129) "There are no standard classifications of probability interpretations, and even the more popular ones may suffer subtle variations from text to text." (p 1130) The classification in this article is representative, as are the authors and ideas claimed for each classification.〕 of probability interpretations which can be called "physical" and "evidential" probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as a die yielding a six) tends to occur at a persistent rate, or "relative frequency", in a long run of trials. Physical probabilities either explain, or are invoked to explain, these stable frequencies. The two main kinds of theory of physical probability are frequentist accounts (such as those of Venn, Reichenbach〔 English translation of the original 1935 German. ASIN: B000R0D5MS〕 and von Mises〔 English translation of the third German edition of 1951 which was published 30 years after the first German edition.〕) and propensity accounts (such as those of Popper, Miller, Giere and Fetzer). Evidential probability, also called Bayesian probability, can be assigned to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility, or the degree to which the statement is supported by the available evidence. On most accounts, evidential probabilities are considered to be degrees of belief, defined in terms of dispositions to gamble at certain odds. The four main evidential interpretations are the classical (e.g. Laplace's)〔 interpretation, the subjective interpretation (de Finetti〔 Translation of the 1937 French original with later notes added.〕 and Savage), the epistemic or inductive interpretation (Ramsey,〔 Contains three chapters (essays) by Ramsey. The electronic version contains only those three.〕 Cox) and the logical interpretation (Keynes and Carnap〔 Carnap coined the notion ''"probability1"'' and ''"probability2"'' for evidential and physical probability, respectively.〕). There are also evidential interpretations of probability covering groups, which are often labelled as 'intersubjective' (proposed by Gillies and Rowbottom〔). Some interpretations of probability are associated with approaches to statistical inference, including theories of estimation and hypothesis testing. The physical interpretation, for example, is taken by followers of "frequentist" statistical methods, such as Ronald Fisher, Jerzy Neyman and Egon Pearson. Statisticians of the opposing Bayesian school typically accept the existence and importance of physical probabilities, but also consider the calculation of evidential probabilities to be both valid and necessary in statistics. This article, however, focuses on the interpretations of probability rather than theories of statistical inference. The terminology of this topic is rather confusing, in part because probabilities are studied within a variety of academic fields. The word "frequentist" is especially tricky. To philosophers it refers to a particular theory of physical probability, one that has more or less been abandoned. To scientists, on the other hand, "frequentist probability" is just another name for physical (or objective) probability. Those who promote Bayesian inference view "frequentist statistics" as an approach to statistical inference that recognises only physical probabilities. Also the word "objective", as applied to probability, sometimes means exactly what "physical" means here, but is also used of evidential probabilities that are fixed by rational constraints, such as logical and epistemic probabilities. ==Philosophy== The philosophy of probability presents problems chiefly in matters of epistemology and the uneasy interface between mathematical concepts and ordinary language as it is used by non-mathematicians. Probability theory is an established field of study in mathematics. It has its origins in correspondence discussing the mathematics of games of chance between Blaise Pascal and Pierre de Fermat in the seventeenth century,〔(Fermat and Pascal on Probability ) (@ socsci.uci.edu)〕 and was formalized and rendered axiomatic as a distinct branch of mathematics by Andrey Kolmogorov in the twentieth century. In its axiomatic form, mathematical statements about probability theory carry the same sort of epistemological confidence shared by other mathematical statements in the philosophy of mathematics.〔Laszlo E. Szabo, ''(A Physicalist Interpretation of Probability )'' (Talk presented on the Philosophy of Science Seminar, Eötvös, Budapest, 8 October 2001.)〕〔Laszlo E. Szabo, Objective probability-like things with and without objective indeterminism, Studies in History and Philosophy of Modern Physics 38 (2007) 626–634 (''(Preprint )'')〕 The mathematical analysis originated in observations of the behaviour of game equipment such as playing cards and dice, which are designed specifically to introduce random and equalized elements; in mathematical terms, they are subjects of indifference. This is not the only way probabilistic statements are used in ordinary human language: when people say that "''it will probably rain''", they typically do not mean that the outcome of rain versus not-rain is a random factor that the odds currently favor; instead, such statements are perhaps better understood as qualifying their expectation of rain with a degree of confidence. Likewise, when it is written that "the most probable explanation" of the name of Ludlow, Massachusetts "is that it was named after Roger Ludlow", what is meant here is not that Roger Ludlow is favored by a random factor, but rather that this is the most plausible explanation of the evidence, which admits other, less likely explanations. Thomas Bayes attempted to provide a logic that could handle varying degrees of confidence; as such, Bayesian probability is an attempt to recast the representation of probabilistic statements as an expression of the degree of confidence by which the beliefs they express are held. Though probability initially had somewhat mundane motivations, its modern influence and use is widespread ranging from Evidence based medicine, through Six sigma, all the way to the Probabilistically checkable proof and the String theory landscape. 〔 (p 1132) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Probability interpretations」の詳細全文を読む スポンサード リンク
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