On Genetic Information, Diversity and Distance

Abstract

General information-theoretic concepts such as f-divergence, f-information and f-entropy are applied to the genetic models where genes are characterized by randomly distributed alleles. The paper thus presents an information-theoretic background for measuring genetic distances between populations, genetic information in various observations on individuals about their alleles and, finally, genetic diversities in various populations. Genetic distances were derived as divergences between frequencies of alleles representing a gene in two different populations. Genetic information was derived as a measure of statistical association between the observations taken on individuals and the alleles of these individuals. Genetic diversities were derived from divergences and information. The concept of genetic f-information introduced in the paper seems to be new. We show that the measures of genetic distance and diversity used in the previous literature are special cases of the genetic f-divergence and f-diversity introduced in the paper and illustrated by examples. We also display intimate connections between the genetic f-information and the genetic f-divergence on one side and genetic f-diversity on the other side. The examples at the same time also illustrate practical computations and applications of the important concepts of quantitative genetics introduced in the paper. We discussed a general class of f- divergence measures that are suitable measures of genetic distance between populations characterized by concrete frequencies of alleles. We have shown that a wide class of genetic information, called f-information, can be obtained from f-divergences and that a wide class of measures of genetic diversity, called f-diversities, can be obtained from the f-divergences and f-information.