Graded comparison of imprecise fitness values

We establish a new way to compare two randomness values of two KBs, which allow us to compare any pair.We compare this new method with the previous ones.We study the behavior for the particular and important cases of uniformity and beta distribution. Genetic algorithms can be used to construct knowledge bases. They are based on the idea of "survival of the fittest" in the same way as natural evolution. Nature chooses the fittest ones in real life. In artificial intelligence we need a method that carries out the comparison and choice. Traditionally, this choice is based on fitness functions. Each alternative or possible solution is given a fitness score. If there is no ambiguity and those scores are numbers, it is easy to order individuals according to those values and determine the fittest ones. However, the process of assessing degrees of optimality usually involves uncertainty or imprecision.In this contribution we discuss the comparison among fitness scores when they are known to be in an interval, but the exact value is not given. Random variables are used to represent fitness values in this situation. Some of the most usual approaches that can be found in the literature for the comparison of those kinds of intervals are the strong dominance and the probabilistic prior method. In this contribution we consider an alternative procedure to order vague fitness values: statistical preference. We first study the connection among the three methods previously mentioned. Despite they appear to be completely different approaches, we will prove some relations among them. We will then focus on statistical preference since it takes into consideration the information about the relation between the fitness values to compare them. We will provide the explicit expression of the probabilistic relation associated to statistical preference when the fitness values are defined by uniform and beta distributions when they are independent, comonotone and countermonotone.