scholarly journals Kingman’s coalescent with erosion

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Félix Foutel-Rodier ◽  
Amaury Lambert ◽  
Emmanuel Schertzer
Keyword(s):  
Kingman’S Coalescent

scholarly journals Computational Inference Beyond Kingman's Coalescent

2015 ◽  
Vol 52 (02) ◽  
pp. 519-537 ◽  
Author(s):  
Jere Koskela ◽  
Paul Jenkins ◽  
Dario Spanò
Keyword(s):  
Importance Sampling ◽  
Genetic Data ◽  
Likelihood Inference ◽  
Data Sets ◽  
Conditional Sampling ◽  
Challenging Problem ◽  
Computational Inference ◽  
Sampling Distributions ◽  
Full Likelihood ◽  
Kingman’S Coalescent

Full likelihood inference under Kingman's coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) methods have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general Λ- and Ξ-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites Λ- and Ξ-coalescents, and use them to obtain ‘approximately optimal’ IS and PAC algorithms for Λ-coalescents, yielding substantial gains in efficiency over existing methods.

scholarly journals Coalescent theory for seed bank models

2001 ◽  
Vol 38 (2) ◽  
pp. 285-300 ◽  
Author(s):  
Ingemar Kaj ◽  
Stephen M. Krone ◽  
Martin Lascoux
Keyword(s):  
Weak Convergence ◽  
Seed Bank ◽  
Temporal Structure ◽  
Seed Banks ◽  
Coalescent Theory ◽  
Egg Banks ◽  
Coalescence Rate ◽  
Age Dependent ◽  
Neutral Mutations ◽  
Kingman’S Coalescent

We study the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations. Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent.

scholarly journals A non-increasing tree growth process for recursive trees and applications

2020 ◽  
pp. 1-26
Author(s):  
Laura Eslava
Keyword(s):  
Tree Growth ◽  
Degree Distribution ◽  
Growth Process ◽  
Convergence Rates ◽  
Positive Probability ◽  
New Process ◽  
Empty Subset ◽  
Recursive Trees ◽  
Stein Method ◽  
Kingman’S Coalescent

Abstract We introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where T n is a rooted labelled tree on n vertices and σ n is a permutation of the vertex labels. The construction of (T n , σ n ) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ T n with positive probability. The key feature of the process is that the shape of T n has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process. We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.

The general coalescent with asynchronous mergers of ancestral lines

1999 ◽  
Vol 36 (4) ◽  
pp. 1116-1125 ◽  
Author(s):  
Serik Sagitov
Keyword(s):  
Population Model ◽  
Marginal Distribution ◽  
Divisible Distribution ◽  
Convergence Criterion ◽  
Positive Probability ◽  
Fixed Size ◽  
Infinitely Divisible ◽  
The Family ◽  
Size Population ◽  
Kingman’S Coalescent

Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral process. We describe a class of asymptotic structures for the ancestral process via a convergence criterion. One of the basic conditions of the criterion prevents simultaneous mergers of ancestral lines. Another key condition implies that the marginal distribution of the family size is attracted by an infinitely divisible distribution. If the latter is normal the coalescent allows only for pairwise mergers (Kingman's coalescent). Otherwise multiple mergers happen with positive probability.

scholarly journals The Genealogy of Samples in Models With Selection

Genetics ◽  
1997 ◽  
Vol 145 (2) ◽  
pp. 519-534 ◽  
Author(s):  
Claudia Neuhauser ◽  
Stephen M Krone
Keyword(s):  
Random Sample ◽  
Dna Sequence ◽  
Selective Advantage ◽  
Allele Frequencies ◽  
The Other ◽  
Rigorous Results ◽  
Haploid Population ◽  
Allele Model ◽  
Neutral Case ◽  
Kingman’S Coalescent

We introduce the genealogy of a random sample of genes taken from a large haploid population that evolves according to random reproduction with selection and mutation. Without selection, the genealogy is described by Kingman's well-known coalescent process. In the selective case, the genealogy of the sample is embedded in a graph with a coalescing and branching structure. We describe this graph, called the ancestral selection graph, and point out differences and similarities with Kingman's coalescent. We present simulations for a two-allele model with symmetric mutation in which one of the alleles has a selective advantage over the other. We find that when the allele frequencies in the population are already in equilibrium, then the genealogy does not differ much from the neutral case. This is supported by rigorous results. Furthermore, we describe the ancestral selection graph for other selective models with finitely many selection classes, such as the K-allele models, infinitely-many-alleles models, DNA sequence models, and infinitely-many-sites models, and briefly discuss the diploid case.

scholarly journals Computational Inference Beyond Kingman's Coalescent

2015 ◽  
Vol 52 (2) ◽  
pp. 519-537 ◽  
Author(s):  
Jere Koskela ◽  
Paul Jenkins ◽  
Dario Spanò
Keyword(s):  
Importance Sampling ◽  
Genetic Data ◽  
Likelihood Inference ◽  
Data Sets ◽  
Conditional Sampling ◽  
Challenging Problem ◽  
Computational Inference ◽  
Sampling Distributions ◽  
Full Likelihood ◽  
Kingman’S Coalescent

Full likelihood inference under Kingman's coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) methods have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general Λ- and Ξ-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites Λ- and Ξ-coalescents, and use them to obtain ‘approximately optimal’ IS and PAC algorithms for Λ-coalescents, yielding substantial gains in efficiency over existing methods.

The general coalescent with asynchronous mergers of ancestral lines

1999 ◽  
Vol 36 (04) ◽  
pp. 1116-1125 ◽  
Author(s):  
Serik Sagitov
Keyword(s):  
Population Model ◽  
Marginal Distribution ◽  
Divisible Distribution ◽  
Convergence Criterion ◽  
Positive Probability ◽  
Fixed Size ◽  
Infinitely Divisible ◽  
The Family ◽  
Size Population ◽  
Kingman’S Coalescent

Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral process. We describe a class of asymptotic structures for the ancestral process via a convergence criterion. One of the basic conditions of the criterion prevents simultaneous mergers of ancestral lines. Another key condition implies that the marginal distribution of the family size is attracted by an infinitely divisible distribution. If the latter is normal the coalescent allows only for pairwise mergers (Kingman's coalescent). Otherwise multiple mergers happen with positive probability.

scholarly journals Gene Genealogies Within a Fixed Pedigree, and the Robustness of Kingman’s Coalescent

Genetics ◽  
2012 ◽  
Vol 190 (4) ◽  
pp. 1433-1445 ◽  
Author(s):  
John Wakeley ◽  
Léandra King ◽  
Bobbi S. Low ◽  
Sohini Ramachandran
Keyword(s):  
Kingman’S Coalescent

scholarly journals Convergence to the coalescent and its relation to the time back to the most recent common ancestor

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Martin Möhle
Keyword(s):  
Overlapping Generations ◽  
Common Ancestor ◽  
Domain Of Attraction ◽  
Population Models ◽  
Recent Common Ancestor ◽  
Most Recent Common Ancestor ◽  
International Audience ◽  
Multiple Collisions ◽  
Full Class ◽  
Kingman’S Coalescent

International audience For the class of haploid exchangeable population models with non-overlapping generations and population size $N$ it is shown that, as $N$ tends to infinity, convergence of the time-scaled ancestral process to Kingman's coalescent and convergence in distribution of the scaled times back to the most recent common ancestor (MRCA) to the corresponding times back to the MRCA of the Kingman coalescent are equivalent. Extensions of this equivalence are derived for exchangeable population models being in the domain of attraction of a coalescent process with multiple collisions. The proofs are based on the property that the total rates of a coalescent with multiple collisions already determine the distribution of the coalescent. It is finally shown that similar results cannot be obtained for the full class of exchangeable coalescents allowing for simultaneous multiple collisions of ancestral lineages, essentially because the total rates do not determine the distribution of a general exchangeable coalescent.

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