By J. F. C. Kingman
Attracts jointly a few mathematical rules which are necessary in inhabitants genetics, focusing on a couple of facets that are either biologically correct and mathematically attention-grabbing.
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Extra resources for Mathematics of Genetic Diversity (CBMS-NSF Regional Conference Series in Applied Mathematics)
Example text
Let us estimate (rather crudely) the probability P that the latter event takes place, so as to get an idea of how vulnerable the original polymorphism is to mutation. 1) indicates that the probability that A0 survives, and takes its place in a new equilibrium, is approximately P = 2e + , where e+ = max(e, 0). Hence we arrive at the approximate formula To see how this probability depends on the parameters of the polymorphism, think of the H' 0j as being independent random variables, drawn from some distribution on (0, 1), and replace P by its expectation/*.
The precise statement is as follows. Let V be the set of all sequences U n ; n ^ 1) satisfying Then for every 6 > 0, there is a probability distribution on V, called the THE NEUTRAL ALTERNATIVE 41 Poisson-Dirichlet distribution with parameter 6, for which the marginal distribution of the first n components is (for any n ^ 1) the limiting joint distribution, as K —> oc and Ka —> 6, of the order statistics p (1) , /? 2). Unfortunately, these marginal distributions (see Watterson [5]) are complicated in form and convey little intuitive idea of the nature of the PoissonDirichlet distribution.
This is a finite-dimensional simplex, whose dimension is one less than the number of partitions of the integer n. 2) defines a linear mapping crn from Hn+1 into H n , thus defining a projective system For such a system, the projective limit is defined as the set of all sequences (/»„; n ^ 1) with and so is just the set FIX of all partition structures. 5) with V replaced by a suitable set of labels for the extreme points of 112. , those that cannot be expressed as mixtures of other partition structures) are those of the formP% for* E V.