scholarly journals Dirichlet approximation of equilibrium distributions in Cannings models with mutation

2017 ◽  
Vol 49 (3) ◽  
pp. 927-959
Author(s):  
Han L. Gan ◽  
Adrian Röllin ◽  
Nathan Ross
Keyword(s):  
Stationary Distribution ◽  
Finite Population ◽  
Dirichlet Distribution ◽  
Finite Size ◽  
First Case ◽  
Probabilistic Description ◽  
New Development ◽  
Equilibrium Distributions ◽  
Haploid Population ◽  
Independent Mutation

AbstractConsider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general. We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite-population stationary distribution for the Wright–Fisher genealogy with general mutation structure and the Cannings exchangeable genealogy with parent independent mutation structure. In the first case, the bound is small if the population is large and the mutations do not depend too much on parent type; 'too much' is naturally quantified by our bound. In the second case, the bound is small if the population is large and the chance of three-mergers in the Cannings genealogy is small relative to the chance of two-mergers; this is the same condition to ensure convergence of the genealogy to Kingman's coalescent. These results follow from a new development of Stein's method for the Dirichlet distribution based on Barbour's generator approach and a probabilistic description of the semigroup of the Wright–Fisher diffusion due to Griffiths and Li (1983) and Tavaré (1984).

AN INVESTIGATION OF THE PHASE TRANSITIONS OF A FAMILY OF PROBABILISTIC AUTOMATA

2004 ◽  
Vol 07 (01) ◽  
pp. 93-123
Author(s):  
HEINZ MÜHLENBEIN ◽  
THOMAS AUS DER FÜNTEN
Keyword(s):  
Phase Transitions ◽  
Stationary Distribution ◽  
Classification Scheme ◽  
Mean Field ◽  
Finite Size ◽  
Order Parameters ◽  
Finite Size Scaling ◽  
Probabilistic Cellular Automata ◽  
Probabilistic Automata ◽  
Major Transitions

We investigate a family of totalistic probabilistic cellular automata (PCA) which depend on three parameters. For the uniform random neighborhood and for the symmetric 1D PCA the exact stationary distribution is computed for all finite n. This result is used to evaluate approximations (uni-variate and bi-variate marginals). It is proven that the uni-variate approximation (also called mean-field) is exact for the uniform random neighborhood PCA. The exact results and the approximations are used to investigate phase transitions. We compare the results of two order parameters, the uni-variate marginal and the normalized entropy. Sometimes different transitions are indicated by the Ehrenfest classification scheme. This result shows the limitations of using just one or two order parameters for detecting and classifying major transitions of the stationary distribution. Furthermore, finite size scaling is investigated. We show that extrapolations to n=∞ from numerical calculations of finite n can be misleading in difficult parameter regions. Here, exact analytical estimates are necessary.

Estimating the frequency of the oldest allele: a bayesian approach

1991 ◽  
Vol 23 (3) ◽  
pp. 456-475 ◽  
Author(s):  
Paul Joyce
Keyword(s):  
Stationary Distribution ◽  
Bayesian Approach ◽  
Dirichlet Distribution ◽  
Allele Frequencies ◽  
Posterior Distributions ◽  
Bayes Estimators ◽  
Infinite Alleles Model

In this paper we calculate posterior distributions associated with a version of the Poisson–Dirichlet distribution called the GEM. The GEM has been shown (by several authors) to be the limiting stationary distribution for allele frequencies listed in age order associated with the neutral infinite alleles model. In view of this result, we use our posterior distributions to calculate Bayes estimators for the frequency of the oldest allele given a sample.

scholarly journals The effect of random selection coefficients on populations of finite size – some particular models

1977 ◽  
Vol 29 (2) ◽  
pp. 97-112 ◽  
Author(s):  
P. J. Avery
Keyword(s):  
Finite Population ◽  
Finite Size ◽  
Random Selection ◽  
Heterozygote Advantage ◽  
Expected Heterozygosity ◽  
Reasonable Degree ◽  
Equal Variance ◽  
The Mean ◽  
Mutant Genes ◽  
Steady State Solutions

SUMMARYThe model of random selection coefficients is considered in the context of a finite population of diploids. The selection coefficients of the homozygotes are allowed to vary with equal variance while the fitness of the heterozygote is kept fixed. Steady-state solutions are found in the case of equal two-way mutation rates with particular reference to the expected heterozygosity. Increasing the variance of the selection coefficients of the homozygotes is found to uniformly increase the heterozygosity for all values of the average selection coefficients and its effect is largest when the selection coefficients of the homozygotes are fully correlated. The fate of mutant genes is also considered in the case of random selection coefficients by looking at the probability of ultimate fixation and the mean times to fixation and extinction. The errors in previous calculations (e.g. Kimura, 1954; Ohta, 1972) are pointed out. It is found that a small average heterozygote advantage together with a reasonable degree of variance in the coefficients can cause an unexpectedly large amount of heterozygosity to be maintained. It is also seen that probabilities of fixation and mean times to boundaries are usually increased by increasing the variance showing that it in fact helps to keep the population heterozygous for much longer than the non-random case. This is in contradiction to some conclusions of Karlin & Levikson (1974) because their haploid results are not easily extendable to the consideration of this sort of diploid model.

Estimating the frequency of the oldest allele: a bayesian approach

1991 ◽  
Vol 23 (03) ◽  
pp. 456-475 ◽  
Author(s):  
Paul Joyce
Keyword(s):  
Stationary Distribution ◽  
Bayesian Approach ◽  
Dirichlet Distribution ◽  
Allele Frequencies ◽  
Posterior Distributions ◽  
Bayes Estimators ◽  
Infinite Alleles Model

In this paper we calculate posterior distributions associated with a version of the Poisson–Dirichlet distribution called the GEM. The GEM has been shown (by several authors) to be the limiting stationary distribution for allele frequencies listed in age order associated with the neutral infinite alleles model. In view of this result, we use our posterior distributions to calculate Bayes estimators for the frequency of the oldest allele given a sample.

scholarly journals Survival of the frequent at finite population size and mutation rate: bridging the gap between quasispecies and monomorphic regimes with a simple model

2018 ◽  
Author(s):  
Bhavin S. Khatri
Keyword(s):  
Population Size ◽  
Stationary Distribution ◽  
Finite Population ◽  
General Description ◽  
Multiple Alleles ◽  
Intermediate Regime ◽  
Theory Of Evolution ◽  
Predictive Theory ◽  
First Time ◽  
Explicit Derivation

In recent years, there has been increased attention on the non-trivial role that genotype-phenotype maps play in the course of evolution, where natural selection acts on phenotypes, but variation arises at the level of mutations. Understanding such mappings is arguably the next missing piece in a fully predictive theory of evolution. Although there are theoretical descriptions of such mappings for the monomorphic (Nμ ≪ 1) and deterministic or very strong mutation (Nμ ⋙ 1) limit, given by developments of Iwasa’s free fitness and quasispecies theories, respectively, there is no general description for the intermediate regime where Nμ ~ 1. In this paper, we address this by transforming Wright’s well-known stationary distribution of genotypes under selection and mutation to give the probability distribution of phenotypes, assuming a general genotype-phenotype map. The resultant distribution shows that the degeneracies of each phenotype appear by weighting the mutation term; this gives rise to a bias towards phenotypes of larger degeneracy analogous to quasispecies theory, but at finite population size. On the other hand we show that as population size is decreased, again phenotypes of higher degeneracy are favoured, which is a finite mutation description of the effect of sequence entropy in the monomorphic limit. We also for the first time (to the author’s knowledge) provide an explicit derivation of Wright’s stationary distribution of the frequencies of multiple alleles.

scholarly journals Rates of convergence to normality for samples from a finite set of random variables

1996 ◽  
Vol 60 (3) ◽  
pp. 355-362 ◽  
Author(s):  
R. D. John ◽  
J. Robinson
Keyword(s):  
Finite Population ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Alternative Proof ◽  
Independent Variables ◽  
First Case ◽  
Finite Set

AbstractRates of convergence to normality of O(N-½) are obtained for a standardized sum of m random variables selected at random from a finite set of N random variables in two cases. In the first case, the sum is randomly normed and the variables are not restricted to being independent. The second case is an alternative proof of a result due to von Bahr, which deals with independent variables. Both results derive from a rate obtained by Höglund in the case of sampling from a finite population.

The infinitely-many-neutral-alleles diffusion model

1981 ◽  
Vol 13 (03) ◽  
pp. 429-452 ◽  
Author(s):  
S. N. Ethier ◽  
Thomas G. Kurtz
Keyword(s):  
Limit Theorem ◽  
Diffusion Process ◽  
Order Statistics ◽  
Diffusion Model ◽  
Stationary Distribution ◽  
Stochastic Models ◽  
Dirichlet Distribution ◽  
Infinite Dimensional ◽  
Image Position ◽  
Discrete Stochastic Models

A diffusion process X(·) in the infinite-dimensional ordered simplex is characterized in terms of the generator defined on an appropriate domain. It is shown that X(·) is the limit in distribution of several sequences of discrete stochastic models of the infinitely-many-neutral-alleles type. It is further shown that X(·) has a unique stationary distribution and is reversible and ergodic. Kingman's limit theorem for the descending order statistics of the symmetric Dirichlet distribution is obtained as a corollary.

scholarly journals Random selective advantages of genes and their probabilities of fixation

1973 ◽  
Vol 21 (3) ◽  
pp. 215-219 ◽  
Author(s):  
Louis Jensen
Keyword(s):  
Finite Population ◽  
Selective Advantage ◽  
Rare Gene ◽  
Haploid Population ◽  
Selection Intensities ◽  
Image Position ◽  
Ultimate Probability ◽  
Random Fluctuations ◽  
Probability Of Fixation

SUMMARYThe question of what is meant by random fluctuations in selection intensities in a finite population is re-examined. The model presented describes the change in the frequency of a gene in a haploid population of size M. It is assumed that in any generation the adaptive values of A and a are equally likely to be 1 + s: 1 or 1: 1 + s. If s is the selective advantage and x the frequency of gene A, then the first two moments of the change in frequency are found to be m(Δx) = x(1 − x)(1 − 2x) θ/2M andwhere E(s2) = θ/M. The ultimate probability of fixation is computed, showing that variability in selection increases the chance of fixation of a rare gene. A more general form for m(Δx) also is obtained. This form is compared with the equation currently used in describing random fluctuations in selection intensities.

scholarly journals Stationary Distribution Markov Chain for COVID-19 Pandemic

2020 ◽  
Vol 1 (2) ◽  
pp. 71-74
Author(s):  
Audi Achmad ◽  
Mahrudinda Mahrudinda ◽  
Budi Ruchjana ◽  
Sudradjat Supian
Keyword(s):  
Markov Chain ◽  
Mathematical Modelling ◽  
Stationary Distribution ◽  
Observation Interval ◽  
Swine Influenza ◽  
New Disease ◽  
Infected People ◽  
First Case ◽  
Long Term Prediction

Coronavirus disease (COVID-19) is a new disease found in the late 2019. The first case was reported on December 31, 2019 in Wuhan, China and spreading all over the countries. The disease was quickly spread to all over the countries. There are 206.900 cases confirmed by March 18, 2020 causing 8.272 death. It was predicted that the number of confirmed cases will continue to increase. On January 30, 2020, WHO declared this as pandemic for the 6th time ever since the swine influenza. There are a lot of researchers which discuss pandemic spreading caused by virus with mathematical modelling. In this paper, we discuss a long-term prediction over the COVID-19 spreading using stationary distribution markov chain. The goal is to analyze the prediction of infected people in long-term by analyzing the COVID-19 daily cases in an observation interval. By analyzing the daily cases of COVID-19 in Indonesia from March 2nd, 2020 until November 1st, 2020, result shown that 53.91% of probability that the COVID-19 daily case will incline in long-term, 44.86% of chance will decline, and 1.23% of chance will stagnant.

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