Generalizing the standard product rule of probability theory and Bayes's Theorem

In this paper the usual product rule of probability theory is generalized by relaxing the assumption that elements of sets are equally likely to be drawn. The need for such a generalization has been noted by Jeffreys [1998. Theory of Probability, 3rd ed, reprinted in Oxford classics Series, Oxford University Press (1st ed.,1939), Oxford. pp. 24–25], among others, in his work on an axiom system for scientific learning from data utilizing Bayes's Theorem. It is shown that by allowing probabilities of elements to be drawn to be different, generalized forms of the product rule and Bayes's Theorem are obtained that reduce to the usual product rule and Bayes's Theorem under certain assumptions that may be satisfactory in many cases encountered in practice in which the principle of insufficient reason is inadequate. Also, in comparing alternative hypotheses, allowing the prior odds to be random rather than fixed provides a useful generalization of the standard posterior odds.