Titlebook: Information Geometry and Population Genetics; The Mathematical Str Julian Hofrichter,Jürgen Jost,Tat Dat Tran Book 2017 Springer Internatio

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,The Wright–Fisher Model,ormulated for diploid populations, and some of the interesting effects occurring in generalizations depend on that diploidy, the formal scheme emerges already for haploid populations. In the basic version, with which we start here, there is a single genetic locus that can be occupied by different al

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Continuous Approximations,y of convergence of Markov processes, see Ethier and Kurtz (Markov processes. Wiley, New York, 1986), but we shall present the proof here in order to keep our treatment self-contained. The proof can be shortened by appealing to general theorems about stochastic processes, but, in order to be self-co

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Recombination,sumed to be linearly ordered. The number of loci, as well as the set of possible alleles for each, are the same for all gametes within the population under consideration. When two gametes are paired, they form an offspring gamete that at some of its sites gets the alleles from the first parent, and

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The Backward Equation,te .(., 0) = .(.) at time . = 0, i.e. having reached the corresponding (generalised) target set. It becomes a parabolic equation upon time reversal, that is, replacing . by − .. We can then treat .(., 0) = .(.) as the initial condition at time . = 0. In view of the biological model behind the Kolmog

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Introduction,Population genetics is concerned with the stochastic dynamics of allele frequencies in a population. In mathematical models, alleles are represented as alternative values at genetic loci.

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Geometric Structures and Information Geometry,We consider the probability simplex

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Moment Generating and Free Energy Functionals,In this section, we will construct the moment generating function for the Wright–Fisher model and derive a partial differential equation that it satisfies. This differential equation encodes all the moment evolution equations from the Sect. .