Theories of Probability

Synopsis

Various definitions of mathematical probability are described and commented upon: (i) the classical formulation, based on the ratio of favourable to total “equally likely” aspects of a system; (ii) formulations based on relative frequency in a sequence of trials, or on the limit of relative frequency as the number of trials is increased indefinitely; (iii) von Mises' refinement of this, in which the sequences obey special postulates ensuring “randomness”; (iv) modern revisions of the classical definition, “equally likely” aspects being replaced by an aggregate, and ratio of aspects by relative measure of a sub-aggregate; (v) the approach of Keynes, in which probability is the “logic of uncertain inference,” developed by a symbolic algebra; (vi) the approach of Jeffreys, by axioms and conventions of procedure, based on the principle of indifference and the use of the rule of inverse probability. It is suggested that, whereas in formulations based on relative frequency, frequency cannot be held to afford any explanation of the facts of frequency, in the a priori formulations, on the other hand, any postulates put forward are still very far from explaining to inquiring minds the commonly observed phenomena of so-called chance. It is further suggested that in actuarial applications the nexus between axioms of probability and causes of mortality is especially obscure, with consequent need of caution in adopting and interpreting the more refined modern methods.