The Axioms of Subjective Probabili

Abstract. This survey recounts contributions to the axiomatic foundations of subjective probability from the pioneering era of Ramsey, de Finetti, Savage, and Koopman to the mid-1980's. It is designed to be accessible to readers who have little prior acquaintance with axiomatics. At the same time, it provides a fairly complete picture of the present state of the measurement-theoretic foundations of subjective probability. Key words and phrases: Axioms for comparative probability, numerical representations, additivity, countable additivity. The probability of any event is the ratio between the value at which an expectation depending on the hap-pening of the event ought to be computed, and the value of the thing expected upon its happening. (Bayes, 1763, page 376) We are driven therefore to the second supposition that the degree of a belief is a causal property of it, which we can express vaguely as the extent to which we are prepared to act on it. (Ramsey, 1931, page 170)... the degree of probability attributed by an indi-vidual to a given event is revealed by the conditions under which he would be disposed to bet on that event. (de Finetti, 1937; from Kyburg and Smokler, 1964, page 101) The intuitive thesis in probability holds that... probability derives directly from the intuition, and is prior to objective experience... (Koopman, 1940a, page 269) Personalistic views hold that probability measures the confidence that a particular individual has in the truth of a particular proposition, for example, the proposi-tion that it will rain tomorrow. (Savage, 1954, page 3) 1