An Introduction to Decision Theory (Cambridge Introductions to Philosophy) by Peterson Martin

Author:Peterson, Martin [Peterson, Martin]
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2009-05-13T22:00:00+00:00

QP 5 needs to be qualified. This axiom does not require that the random variable in question really exists. It is sufficient that the agent believes that this random variable exists. De Groot’s approach to subjective probability theory makes no assumption about the nature of the external world – all that matters is the structure of internal subjective beliefs. QP 5 is thus consistent with the world being deterministic.

To understand what work is carried out by QP 5, suppose an agent wishes to determine her subjective probability for the two events ‘rain here within the hour’ and ‘no rain here within the hour’. Then, since the set of events E only contains two elements, it is not possible to obtain a quantitative probability function by only comparing those two events. The set of events has to be extended in some way. QP 5 is the key to this extension. In a uniform probability distribution all elements (values) are equally likely. As an example, think of a roulette wheel in which the original numbers have been replaced with an infinite number of points in the interval [0, 1]. Then, by applying QP 5, the set of events can be extended to the union of the two original events and the infinite set of events ‘the wheel stops at x (0 x 1)’, etc.

Theorem 7.3

QP 1–5 are jointly sufficient and necessary for the existence of a unique function p that assigns a real number in the interval [0, 1] to all elements in E, such that X Y if and only if p(X) p(Y). In addition, p satisfies Kolmogorov’s axioms (see page 120).

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