The definition of conditional probabilities is based upon the existenc
e of a joint probability, However, a reconstruction of the joint proba
bility from given conditional probabilities imposes certain constraint
s upon the latter, so that if several conditional probabilities are ch
osen arbitrarily, the corresponding joint probability may not exist. S
uch an incompleteness in conditional probabilities can be eliminated b
y introducing complex probabilities. The physical meaning of the new m
athematical formalism, as well as its relation to quantum probabilitie
s, is discussed. (C) 1998 Elsevier Science B.V.