The corresponding F ST measures and their various generalizations for subdivided populations have also played a central role in population genetics and evolutionary biology. Genetic analysis of populations has nearly always relied on measures based on expected heterozygosities or gene identities, because these link to variance and the binary nature of sexual reproduction and diploid inheritance. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are creditedĭata Availability: All relevant data are within the paper and its Supporting Information files.įunding: The Mathematics Research Center (of Taiwan Ministry of Science and Technology), The Population Biology Foundation, and the Ministry of Science and Technology, Taiwan, Contract 100-2118-M007-006-MY3 ( ).

Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.Ĭitation: Chao A, Jost L, Hsieh TC, Ma KH, Sherwin WB, Rollins LA (2015) Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model. We apply our measures to data from the common starling ( Sturnus vulgaris) in Australia. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also identify a bridge between the two models of mutation. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions.

We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Introduced by Kullback and Leiber in 1951 ( ).Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications (2) these measures connect directly to the rich predictive mathematics of information theory (3) Shannon entropy is completely additive and has an explicitly hierarchical nature and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. Relative entropy, also known as the Kullback-Leibler divergence, was relative_entropy ( p, q, b=2.0 ) ¶Ĭompute the base- b relative entropy between posterior ( p) and prior The conditional mutual information Return typeįloat pyinform.shannon. xyz = Dist () > xz = Dist () > yz = Dist () > z = Dist () > shannon.