Foundations for conditional probability

Ladislav Mečíř

Published 2019-07-08Version 1

We analyze several formalizations of conditional probability and find a new one that encompasses all. Our main result is that a preference relation on random quantities called a plausible preorder induces a coherent conditional expectation; and vice versa, that every coherent function can be extended to a conditional expectation induced by a plausible preorder. The advantages of our approach include a convenient justification of probability laws by the properties of plausible preorders, independence on probability interpretations, or the ability to extend conditional probability to any nonzero condition. In particular, if C is a nonzero condition and \Prob is coherent, then it can be extended so that \Prob(0|C)=0, \Prob(C|C)=1 and \Prob(1|C)=1, no matter whether \Prob(C) is zero or whether it is defined.