ダッチ・ブック論証による信念の度合いとしての確率解釈
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概要
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研究ノート1. 導入2. 主観説とダッチ・ブックの論証 2.1. 主観説の方策 : 信念の度合いから確率への2段階の置き換え 2.2. 賭けの一般形式と賭け指数 2.3. 第1段階 : 信念の度合いと公平な賭け指数 2.4. 第2段階 : 公平な賭け指数と確率 ダッチ・ブック定理3. 結論 : 第1段階と第2段階の統合付録 ダッチ・ブック定理の証明Lots of efforts have been paid for interpreting the concept of probability by another familiar concept such as ignorance, degree of a partial logical entailment, degree of belief, frequency, and propensity. In this paper the subjective theory is addressed. According to this theory, probability is interpreted as coherent degree of belief of a particular individual. This interpretation is achieved through following the two-step replacements: (1) Degree of belief is interpreted as fair betting quotient; (2) Fair betting quotient is interpreted as probability. The first replacement is based on the claim that in a bet (decision-making in an uncertain situation) a bettor's degree of belief whether an event will occur can be measured by a real number which she gives through her judgement on the fairness of the bet. The second replacement is based on the fact that when a bettor makes bets on events, in order to be guaranteed not to lose whatever happens (in order to be coherent) she should assign her betting quotients in accordance with the probability axioms, and vice versa. It is the so-called Dutch Book Theorem that guarantees this fact mathematically. The purpose of this paper is to clarify and confirm the contents of the subjective theory in terms of betting systems along the following approach (Ramsey (1931), de Finetti (1937), Howson Urbach (1993), Gillies (2000), etc.).
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