Combining Hints

Abstract

If there are several hints relative to the same question and hence relative to the same frame of discernment Ө, then the problem of combining the information contained in each hint arises. This is the very basic operation to be defined in the theory of hints. Its intuitive idea is very simple and logically follows from the semantics given to the four elements defining a hint. Recall that for each hint to be combined there is exactly one correct interpretation and the answer to the question considered necessarily lies in its corresponding focal set. The vector formed by these correct interpretations is called the correct combined interpretation. Whereas there are several vectors of interpretations (combined interpretations) which are obtained by selecting an interpretation for each hint, there is exactly one correct combined interpretation. Then, of course, the answer to the question considered must be in the intersection of the focal sets associated with the individual interpretations forming this combined interpretation. Therefore, a combined interpretation leading to an empty intersection is certainly not correct and it is called contradictory. Contradictory interpretations represent a new information for the determination of the correct combined interpretation. This information must be integrated in the original knowledge available for this question, knowledge which is described by the probability that an arbitrary combined interpretation is correct. Of course, the integration is realized by conditioning this probability on non-contradictory interpretations.