Abstract: The concept of entropy is expanded from real number field to complex num-ber field, and a complete complex entropy definition is given for the first time. The complex entropy will be in agreement with classical entropy, if the probability in it takes the traditional value. Therefore, the complex entropy contains the classical entropy. It is the expansion of a classical one. On the other hand, a new measure of Bayes-E decision-making under risk is defined by use of complex entropy. This new measure is called transference entropy, and its effectiveness has be proved as well. The uncompleteness of Bayes decision analysis method has been overcome, and a new decision programme-Bayes-E decision method is obtained. The usage of Bayes-E decision method is explained with the practical decision problem as an example.

Key words: complex entropy, transference entropy, Bayes-E decision procedure