Population genetics and Markov chains

1988 ◽  
pp. 148-182
Author(s):  
Henry C. Tuckwell
Keyword(s):  
Population Genetics ◽  
Markov Chains

An extension of a convergence theorem for Markov chains arising in population genetics

2016 ◽  
Vol 53 (3) ◽  
pp. 953-956 ◽  
Author(s):  
Martin Möhle ◽  
Morihiro Notohara
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Convergence Theorem ◽  
Transition Matrix ◽  
Initial Distribution ◽  
Generator Matrix ◽  
Finite Dimensional ◽  
Finite State ◽  
Initial Distributions ◽  
Finite State Space

AbstractAn extension of a convergence theorem for sequences of Markov chains is derived. For every positive integer N let (XN(r))r be a Markov chain with the same finite state space S and transition matrix ΠN=I+dNBN, where I is the unit matrix, Q a generator matrix, (BN)N a sequence of matrices, limN℩∞cN= limN→∞dN=0 and limN→∞cN∕dN=0. Suppose that the limits P≔limm→∞(I+dNQ)m and G≔limN→∞PBNP exist. If the sequence of initial distributions PXN(0) converges weakly to some probability measure μ, then the finite-dimensional distributions of (XN([t∕cN))t≥0 converge to those of the Markov process (Xt)t≥0 with initial distribution μ, transition matrix PetG and limN→∞(I+dNQ+cNBN)[t∕cN]

Time reversal and age distributions, I. Discrete-time Markov chains

1980 ◽  
Vol 17 (1) ◽  
pp. 33-46 ◽  
Author(s):  
S. Tavaré
Keyword(s):  
Population Genetics ◽  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Time Reversal ◽  
Age Distribution ◽  
Applied Probability ◽  
Discrete Time Markov Chain

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

scholarly journals Large Deviation Results for a Class of Markov Chains Arising from Population Genetics

1989 ◽  
Vol 17 (3) ◽  
pp. 1124-1146 ◽  
Author(s):  
Gregory J. Morrow ◽  
Stanley Sawyer
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Large Deviation

A convergence theorem for markov chains arising in population genetics and the coalescent with selfing

1998 ◽  
Vol 30 (2) ◽  
pp. 493-512 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Population Size ◽  
Mutation Rate ◽  
Convergence Theorem ◽  
Population Models ◽  
Fisher Model

A simple convergence theorem for sequences of Markov chains is presented in order to derive new ‘convergence-to-the-coalescent’ results for diploid neutral population models.For the so-called diploid Wright-Fisher model with selfing probability s and mutation rate θ, it is shown that the ancestral structure of n sampled genes can be treated in the framework of an n-coalescent with mutation rate ̃θ := θ(1-s/2), if the population size N is large and if the time is measured in units of (2-s)N generations.

Estimating the Ratios of the Stationary Distribution Values for Markov Chains Modeling Evolutionary Algorithms

2009 ◽  
Vol 17 (3) ◽  
pp. 343-377 ◽  
Author(s):  
Boris Mitavskiy ◽  
Chris Cannings
Keyword(s):  
Stochastic Process ◽  
Population Genetics ◽  
Markov Chain ◽  
Evolutionary Algorithms ◽  
Markov Chains ◽  
Evolutionary Algorithm ◽  
Stationary Distribution ◽  
Genetic Load ◽  
Construction Method ◽  
Expected Time

The evolutionary algorithm stochastic process is well-known to be Markovian. These have been under investigation in much of the theoretical evolutionary computing research. When the mutation rate is positive, the Markov chain modeling of an evolutionary algorithm is irreducible and, therefore, has a unique stationary distribution. Rather little is known about the stationary distribution. In fact, the only quantitative facts established so far tell us that the stationary distributions of Markov chains modeling evolutionary algorithms concentrate on uniform populations (i.e., those populations consisting of a repeated copy of the same individual). At the same time, knowing the stationary distribution may provide some information about the expected time it takes for the algorithm to reach a certain solution, assessment of the biases due to recombination and selection, and is of importance in population genetics to assess what is called a “genetic load” (see the introduction for more details). In the recent joint works of the first author, some bounds have been established on the rates at which the stationary distribution concentrates on the uniform populations. The primary tool used in these papers is the “quotient construction” method. It turns out that the quotient construction method can be exploited to derive much more informative bounds on ratios of the stationary distribution values of various subsets of the state space. In fact, some of the bounds obtained in the current work are expressed in terms of the parameters involved in all the three main stages of an evolutionary algorithm: namely, selection, recombination, and mutation.

Lines-of-descent and genealogical processes, and their applications in population genetics models

1984 ◽  
Vol 16 (1) ◽  
pp. 27-27 ◽  
Author(s):  
Simon Tavaré
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Time Scale ◽  
Diffusion Time ◽  
Single Locus ◽  
Diffusion Approximations ◽  
Central Importance ◽  
Haploid Population ◽  
Lines Of Descent

This paper reviews a variety of results for genealogical (or line-of-descent) processes that arise in connection with the theory of some classical selectively neutral haploid population genetics models. While some new results and derivations are included, the principal aim of the paper is to demonstrate the central importance and simplicity of genealogical Markov chains in this theory. Considerable attention is given to ‘diffusion time scale’ approximations of such genealogical processes. A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach. Other examples where the genealogical process plays an explicit role (for example, the infinite-sites models) are discussed.

A convergence theorem for markov chains arising in population genetics and the coalescent with selfing

1998 ◽  
Vol 30 (02) ◽  
pp. 493-512 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Population Genetics ◽  
Markov Chains ◽  
Population Size ◽  
Mutation Rate ◽  
Convergence Theorem ◽  
Population Models ◽  
Fisher Model

A simple convergence theorem for sequences of Markov chains is presented in order to derive new ‘convergence-to-the-coalescent’ results for diploid neutral population models. For the so-called diploid Wright-Fisher model with selfing probability s and mutation rate θ, it is shown that the ancestral structure of n sampled genes can be treated in the framework of an n-coalescent with mutation rate ̃θ := θ(1-s/2), if the population size N is large and if the time is measured in units of (2-s)N generations.

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