Characterizing the principle of minimum cross-entropy within a conditional-logical framework

The principle of minimum cross-entropy (ME-principle) is often used as an elegant and powerful tool to build up complete probability distributions when only partial knowledge is available. The inputs it may be applied to are a prior distribution P and some new information R, and it yields as a result the one distribution P* that satisfies R and is closest to P in an information-theoretic sense. More generally, it provides a "best" solution to the problem "How to adjust P to R?" In this paper, we show how probabilistic conditionals allow a new and constructive approach to this important principle. Though popular and widely used for knowledge representation, conditionals quantified by probabilities are not easily dealt with. We develop four principles that describe their handling in a reasonable and consistent way, taking into consideration the conditional-logical as well as the numerical and probabilistic aspects. Finally, the ME-principle turns out to be the only method for adjusting a prior distribution to new conditional information that obeys all these principles. Thus a characterization of the ME-principle within a conditional-logical framework is achieved, and its implicit logical mechanisms are revealed clearly. (C) 1998 Elsevier Science B.V.