Coherence and extensions of stochastic matrices

In this paper a review of some general results on coherence of conditional probability assessments is given. Then, a necessary and sufficient condition on coherence of two finite families of discrete conditianal probability distributions, represented by two stochastic matrices P and Q, is obtained. Moreover, the possible extensions of the assessment (P,Q) to the marginal distributions are examined and explicit formulas for them are given in some special case. Finally, a general algorithm to check coherence of (P,Q) and to derive its extensions is proposed.