|
|
|
|
|
|
|
where b = s1/(µ1 - µr). Accordingly, |
|
|
|
|
|
|
|
|
4. Application to Schemata |
|
|
|
|
|
|
|
|
We are ready now to apply the criterion just developed to the general problem of ranking schemata. The basic problem was rephrased as one of minimizing the performance losses inevitably coupled with any attempt to increase confidence in an observed ranking of schemata. The theorem just proved provides a guideline for solving this problem by indicating how trials should be allocated among the schemata of interest. |
|
|
|
|
|
|
|
|
To see this note first that the central limit theorem, used at the heart of the proof of Theorem 5.3, applies to any sequence of independent random variables having means and variances. As such it applies to the observed average payoff  of a sequence of trials of the schema x under any probability distribution P over  (cf. chapter 4). It even applies when the distribution over a changes with time (a fact we will take advantage of with reproductive plans). In particular, then, Theorem 5.3 applies to any given set of r schemata. It indicates that under a good adaptive plan the number of trials of the (observed) best will increase exponentially relative to the total number of trials allocated to the remainder. |
|
|
|
|
|
|
|
|
Near the end of chapter 4 it was proposed that the observed performance rankings of schemata be stored by selecting an appropriate (small) set of elements  from so that the rank of each schema would be indicated by the relative number of instances of x in . Theorem 5.3 suggests an approach to developing , or rather a sequence  ,according to the sequence of observations of schemata. Let the number of instances of x in the set  represent the number of observations of x at time t. Then the number of instances of x in the set  represents the total number of observations of x through time T. If schema x should persist as the observed best, Theorem 5.3 indicates that x's portion of should increase exponentially with respect to the remainder. We can look at this in a more "instantaneous" sense. x's portion of corresponds to the rate at which x is being observed, i.e., to the "derivative" of the function giving x's increase. Since the derivative of an exponential is an exponential, it seems natural to have x's portion Mx(t)of increase exponentially with t (at least until x |
|
|
|
|
|
