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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Lambda-coalescents and population geneticinference
Matthias {Birkner1, Steinrucken2}1University of Munich and 2TU Berlin
Joint project with
Jochen Blath2
Eindhoven, 25th/27th March 2009
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Outline
Introduction
Beta(2 − α,α)-coalescents
Dawson-Watanabe and Fleming-Viot processes, time-changerelation
Mutation models: Infinitely-many-alleles, infinitely-many-sites
Computing likelihoods under Λ-coalescents
Importance sampling methods
Summary & Outlook
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Genetic variability at the mitochondrial cyt b locus in
Atlantic cod
468 481 487 488 490 496 508 523 562 601 631 643 649 685 691
66 t a a c a a t g a t g a c c g
17 - - - - - - c - - - - - - - -
14 - - - - - - - a - - - - - t -
8 - - - - - - - - - - - - - - t
1 - - - - - - - - - - - - - t -
2 - - - t - - - - - - - - - - -
1 - - - - - - - a - - - g - t -
1 - - - - - - - - - - - - t - -
1 - - - - g - c - - - - - - - -
1 - - - - - g - - - - - - - - -
1 - - g - - - - - - - a - - - t
1 - - - - - - c - g - - - - - -
1 g - - - - - - - - - - - - - -
1 - - - - - - - - - c - - - - -
1 - c - - - - c - - - - - - - -
(a random subsample of the sample described in Arnason, Genetics 2004)
![Page 4: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/4.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
The Great Obsession of population geneticists (J. Gillespie)
What evolutionary forces could have lead to such
divergence between individuals of the same species?
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
The Great Obsession of population geneticists (J. Gillespie)
What evolutionary forces could have lead to such
divergence between individuals of the same species?
In this talk, we will focus on neutral genetic variation, and thus theinterplay of mutation and genetic drift.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Wright-Fisher model: The fundamental model for ‘genetic drift’
A (haploid) population of N individuals per generation,each individual in the present generation picks a ‘parent’ atrandom from the previous generation,genetic types are inherited (possibly with a small probability ofmutation).
past
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Wright-Fisher model: The fundamental model for ‘genetic drift’
A (haploid) population of N individuals per generation,each individual in the present generation picks a ‘parent’ atrandom from the previous generation,genetic types are inherited (possibly with a small probability ofmutation).
past
![Page 8: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/8.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Wright-Fisher model: The fundamental model for ‘genetic drift’
A (haploid) population of N individuals per generation,each individual in the present generation picks a ‘parent’ atrandom from the previous generation,genetic types are inherited (possibly with a small probability ofmutation).
past
![Page 9: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/9.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Wright-Fisher model: The fundamental model for ‘genetic drift’
A (haploid) population of N individuals per generation,each individual in the present generation picks a ‘parent’ atrandom from the previous generation,genetic types are inherited (possibly with a small probability ofmutation).
past
![Page 10: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/10.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Genealogical point of view
Sample n (≪ N) individuals from the ‘present generation’
past
present
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Kingman’s coalescent
Theorem (Kingman (& Hudson, Griffiths, ...), 1982)
In the limit N → ∞, the genealogy of an n-sample, measured in units of N generations, isdescribed by a continuous-time Markov chainwhere each pair of lineages merges at rate 1.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Kingman’s coalescent
Theorem (Kingman (& Hudson, Griffiths, ...), 1982)
In the limit N → ∞, the genealogy of an n-sample, measured in units of N generations, isdescribed by a continuous-time Markov chainwhere each pair of lineages merges at rate 1.
Robustness. The same limit appears for any exchangeable offspringvectors
(ν1, . . . , νN), (independent over generations),
if time is measured in units ofN
σ2generations, where
σ2 = limN→∞
Var(ν1)
(under a third moment condition on ν1).
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Modeling neutral variation: Superimposing types on the
coalescent
Assume that the considered genetic typesdo not affect their bearer’s reproductive succes.
If as population size N → ∞,N
σ2× mutation prob. per ind. per generation → r ,
the type configuration in the sample can be described by puttingmutations with rate r along the genealogy.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Modeling neutral variation: Superimposing types on the
coalescent
Assume that the considered genetic typesdo not affect their bearer’s reproductive succes.
If as population size N → ∞,N
σ2× mutation prob. per ind. per generation → r ,
the type configuration in the sample can be described by puttingmutations with rate r along the genealogy.
Kingman’s coalescent is the standard model of mathematicalpopulation genetics.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Question
What if the variability of offspring numbers across individuals is solarge that reasonably
σ2 ≈ ∞ ?
This might happen e.g. in marine species (so-called reproduction
sweepstakes).
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Coalescents with multiple collisions, aka ‘Λ-coalescents’
While n lineages, any k coalesce at rate
λn,k =
∫
[0,1]xk−2(1 − x)n−k Λ(dx), where Λ is a finite measure on
[0, 1]. (Sagitov, 1999; Pitman, 1999).
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Coalescents with multiple collisions, aka ‘Λ-coalescents’
While n lineages, any k coalesce at rate
λn,k =
∫
[0,1]xk−2(1 − x)n−k Λ(dx), where Λ is a finite measure on
[0, 1]. (Sagitov, 1999; Pitman, 1999).
Interpretation:re-write λn,k =
∫[0,1] x
k(1 − x)n−k 1x2 Λ(dx) to see:
at rate 1x2 Λ([x , x + dx ]), an ‘x-resampling event’ occurs.
Thinking forwards in time, this corresponds to an event in whichthe fraction x of the total population is replaced by the offspring ofa single individual.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Coalescents with multiple collisions, aka ‘Λ-coalescents’
While n lineages, any k coalesce at rate
λn,k =
∫
[0,1]xk−2(1 − x)n−k Λ(dx), where Λ is a finite measure on
[0, 1]. (Sagitov, 1999; Pitman, 1999).
Interpretation:re-write λn,k =
∫[0,1] x
k(1 − x)n−k 1x2 Λ(dx) to see:
at rate 1x2 Λ([x , x + dx ]), an ‘x-resampling event’ occurs.
Thinking forwards in time, this corresponds to an event in whichthe fraction x of the total population is replaced by the offspring ofa single individual.
Form of rates stems from λn,k = λn+1,k + λn+1,k+1 (consistencycondition).
![Page 19: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/19.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Coalescents with multiple collisions, aka ‘Λ-coalescents’
While n lineages, any k coalesce at rate
λn,k =
∫
[0,1]xk−2(1 − x)n−k Λ(dx), where Λ is a finite measure on
[0, 1]. (Sagitov, 1999; Pitman, 1999).
Interpretation:re-write λn,k =
∫[0,1] x
k(1 − x)n−k 1x2 Λ(dx) to see:
at rate 1x2 Λ([x , x + dx ]), an ‘x-resampling event’ occurs.
Thinking forwards in time, this corresponds to an event in whichthe fraction x of the total population is replaced by the offspring ofa single individual.
Form of rates stems from λn,k = λn+1,k + λn+1,k+1 (consistencycondition).Note: Λ = δ0 corresponds to Kingman’s coalescent.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Cannings’ models in the
‘domain of attraction of a Λ-coalescent’
Fixed population size N, exchangeable offspring numbers in onegeneration (
ν1, ν2, . . . , νN
).
Sagitov (1999), Mohle & Sagitov (2001) clarify under whichconditions the genealogies of a sequence of exchangeable finitepopulation models are described by a Λ-coalescent:
cN := pair coalescence probability over one generation → 0
( cN = 1N−1E[ν1(ν1 − 1)] )
two double mergers asymptotically negligible compared to onetriple merger
NcN Pr(a given family has size ≥ Nx
)∼
∫ 1x
y−2Λ(dy)
Time is measured in 1/cN generations (in general 6= 1/pop. size)
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Note: There are many Λ-coalescents.
Maybe a natural “first candidate”:
Λ = wδ0 + (1 − w)δψ with w , ψ ∈ (0, 1)
(as considered by Eldon & Wakeley, Genetics 2006)
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
A ‘heavy-tailed’ Cannings model and Beta-coalescents
Haploid population of size N. Individual i has Xi potential
offspring, X1,X2, . . . ,XN are i.i.d. with mean m := E[X1
]> 1,
Pr(X1 ≥ k
)∼ Const. × k−α with α ∈ (1, 2).
Note: infinite variance.
Sample N without replacement from all potential offspring to formthe next generation.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
A ‘heavy-tailed’ Cannings model and Beta-coalescents
Haploid population of size N. Individual i has Xi potential
offspring, X1,X2, . . . ,XN are i.i.d. with mean m := E[X1
]> 1,
Pr(X1 ≥ k
)∼ Const. × k−α with α ∈ (1, 2).
Note: infinite variance.
Sample N without replacement from all potential offspring to formthe next generation.
Theorem (Schweinsberg, 2003)Let cN = prob. of pair coalescence one generation back in N-thmodel.cN ∼ const. N1−α, measured in units of 1/cN generations, thegenealogy of a sample from the N-th model is approximatelydescribed by a Λ-coalescent with Λ = Beta(2 − α,α).(
Beta(2 − α,α)(dx) = 1[0,1](x) 1Γ(2−α)Γ(α) x
1−α(1 − x)α−1 dx)
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Why Λ = Beta(2 − α, α)?
Heuristic argument:
Probability that first individual’s offspring provides
more than fraction y of the next generation,
given that the family is substantial (i.e. given X1 ≥ εN, for y > ε)
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
Why Λ = Beta(2 − α, α)?
Heuristic argument:
Probability that first individual’s offspring provides
more than fraction y of the next generation,
given that the family is substantial (i.e. given X1 ≥ εN, for y > ε)
≈ P
( X1
X1 + (N − 1)m≥ y
∣∣∣X1 ≥ εN)
= P
(X1 ≥ (N − 1)m
y
1 − y
∣∣∣X1 ≥ εN)
∼ const.(1 − y)α
yα= const.’ Beta(2 − α,α)([y , 1]).
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
The Great ObsessionWright-Fisher model and Kingman’s coalescentMultiple merger coalescentsThe Beta(2 − α, α)-class
The family Beta(2 − α, α), α ∈ (1, 2]
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
1.91.51.1
Kingman’s coalescent includedas boundary case:Beta(2 − α,α) → δ0 weakly as α→ 2.
Smaller α means tendency towardsmore extreme resampling events.
For α ≤ 1, corresponding coalescentsdo not come down from infinity.
Beta(2 − α,α)-coalescents appear as genealogies of α-stablecontinuous mass branching process (via a time-change).
Scaling relation of mutation rate per generation relative topopulation size depends on α!
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Free branching: Galton-Watson processes
individuals have a random number of offspring, independently,according to a fixed probability distribution
the totality of offspring forms the next generation
da capo ...
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
![Page 29: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/29.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
![Page 30: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/30.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
![Page 31: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/31.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
![Page 32: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/32.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Galton-Watson forests (with ordered individuals)tim
e
ξt(n) := # descendants of founders 1, . . . , n alive in generation t .For fixed n, ξt(n), t ∈ Z+ is a Galton-Watson process, ξ0(n) = n.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Free branching & neutral typestim
e
Example: there are 2 neutral types, red and blue.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Fixed population size: Cannings models (reprise)
fixed population size N
individuals have a random number of offspring,the offspring numbers (ν1, . . . , νN) are exchangeable with∑N
i=1 νi = N
the totality of offspring forms the next generation
da capo ...
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings modeltim
e
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Genealogies in a Cannings model: Adding typestim
e
We can again think of neutral types (for the moment, withoutmutation), e.g. red and blue.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Question
Relation between
and ?
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Relations?
A first answer
Condition a Galton-Watson process to have constant
population size N, obtain a model from Cannings’ class.
Easy, but is there more?
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Relations?
In a Cannings model, we think ofrelative frequencies of types.
We can do the same in a free branching model by normalising withthe current total population size (at least before extinction), i.e. byconsidering
ξt(n) :=ξt(n)
ξt(N).
/ ≈ ?
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Relations?
Yes, in a suitable limit of population size N → ∞.
20 40 60 80 100 120
20
40
60
80
100
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Continuous state branching processes (Jirina, Silverstein, Lamperti, ... )
Z (N) = (Z(N)k )k=0,1,2,..., N ∈ N a sequence of Galton-Watson
processes (possibly with offspring distributions depending on N),
mN → 0 mass rescaling,
Z(N)0 = [m−1
N ] ( + conditions ... ).
(mNZ
(N)[Nt]
)t≥0
=⇒ X ,
where X is a continuous state branching process, i.e. an R+-valuedMarkov process that enjoys the branching property:
X ,X ′,X ′′ independent copies, X0 = x ,X ′0 = x ′,X ′′
0 = x ′′, x = x ′ + x ′′
=⇒ Xlaw= X ′ + X ′′.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Continuous state branching processes: Favourite example
Assume the N-th offspring distribution has
mean = 1 + µ/N + o(1/N),
variance = σ2 + o(1)
(and, say, uniformly bounded third moment).
Z(N)0 = N, then
(N−1Z
(N)[Nt]
)t≥0
⇒ Feller’s branching diffusion,
generator
L(2)f (z) =1
2σ2zf ′′(z) + µzf ′(z).
(i.e. Var[Zt+∆t −Zt|Zt ] ≈ σ2Zt∆t, E[Zt+∆t −Zt|Zt ] ≈ µZt∆t.)
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Continuous state branching processes: stable case, α < 2
If the approximating offspring distributions are in the domain ofattraction of a stable law of index α ∈ (0, 2), i.e.
P(more than n children) ∼ Const. × n−α
(note: in particular, no variance),the limit process Z will have discontinuous paths. Generator
L(α)f (z) = cαzf ′(z)+z
∫
(0,∞)
{f (z+h)−f (z)−h1(0,1](h)f ′(z)
}h−1−αdh.
Interpretation:if the present mass is z , a new litter of mass h is produced at ratez × h−1−α.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Stable CSBPes, some properties
Why the name?
Lamperti’s construction connects them with stable Levy
processes: Xt = Y(∫ t
0 Xs ds), where Y is a stable process
without negative jumps, stopped upon hitting 0.
Scaling properties
Equation for Laplace transforms
Existence of mean:α > 1 ⇒ Ex Xt <∞,α < 1 ⇒ Ex Xt = ∞ for all t > 0.
Extinction/Explosion:α < 1 ⇒ X has growing paths, explodes in finite time,α > 1 ⇒ X becomes extinct in finite time.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Dawson-Watanabe superprocesses
For our purposes here: the continuous mass & timeanalogue of the “ragged boundary” picture:
Z a given CSBP. The corresponding Dawson-Watanabe
superprocess∗ is a process X with values in the measures on [0, 1]such that
for B ⊂ [0, 1]: (Xt(B))t≥0law= Z , started from Z0 = |B |,
X·(B1),X·(B2), . . . ,X·(Bn) are independent if B1, . . .Bn aredisjoint.
Interpretation: Xt(B) = mass of descendants alive at time t
whose ancestors where ∈ B∗Note: this is a “toy version”.
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Λ-Fleming-Viot processes (Donnelly & Kurtz, Bertoin & Le Gall)
For our purposes here: the continuous mass & timeanalogue of the “straight boundary” picture:
(ρt)t≥0 a Markov process with values in the probability measureson [0, 1]. Interpretation (if ρ0 = uniform measure):
ρt(B) =fraction of mass alive at time t whose ancestors attime 0 where in B ⊂ [0, 1].
On test functions of the formF (ρ) =
∫. . .
∫ρ(dx1) . . . ρ(dxp)f (x1, . . . , xp), the generator acts as
LFV ,ΛF (ρ) =∑
J⊂{1,...,p},
|J|≥2
λp,|J|
∫. . .
∫ρ(dx1) . . . ρ(dxp)
(f (aJ
1 , . . . , aJp) − f (a1, . . . , ap)
).
↑aJi = amin J if i ∈ J
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IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Λ-FV processes: Interpretation of “non-Kingman” part
N a Poisson point process on [0,∞) × (0, 1] with intensitymeasure dt ⊗ y−2Λ(dy).
0 20 40 60 80 100
0.00.2
0.40.6
0.81.0
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·
· ··· · ·· ·· ·
·
·· · ·
·
· ·
·
· ·
·
· ····
· ··
· ·
·
·
·
·
· ···· ·· ·· ·
·
·
·
·· ·
·
·· ·· ·· ·· · · · ·· · ·
·
··
··· ·
·
·
·
·
·
··
·· ··
·
··· · ·· ··
·
· ·
·
·· ·
·
·· · ··· ··
·· ···
·
···
· ···· ·
···
··
·· ·· ·· ·
·
·
·
··
·
·· ·· · ·
·
·· ·
·
·
· ·
· ·
· ··
·
· ··
···
·
·
·
···
·
·· · ···
·
· · ··
·
· ·· · ·· · ··
·
··· ·· ·
·
·· · ·· ·· ·
·
· · ····
· ···
·
· ··· ··
·
···
·
·· ·· · ··· ·· ··
·
··· ··· ·· ·
··
· · ·· ···
·
· ·· ··· ·· · · · ·
·
·
·
···
·
·· ·
y
time
If (t, y) is a point of N , an individual X is chosen according to thecurrent ρ(t−), and ρ→ yδX + (1 − y)ρ.
Note: alternative form of generator
LFV ,λG(ρ) =∫
y−2Λ(dy)∫ρ(dx)
(G(yδx + (1 − y)ρ) − G(ρ)
)
![Page 67: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/67.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Duality between coalescent and FV
For a partition γ of {1, . . . , p} with |γ| classes and a probabilitymeasure ρ on [0, 1] consider test functions of the type
Φf (γ, ρ) :=
∫ρ(dx1) . . . ρ(dx|γ|)f
(~y(γ; x1, . . . , x|γ|)
)
↑yj = xi if j ∈ i -th class of γ
(“assign an independent ρ-sample to each class of γ”), wheref : [0, 1]p → R.
Then we have with(Γ
(p)t ) . . . . . . . . restriction of Λ-coalescent to {1, . . . , p},
(ρt) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Λ-FV process
E
[Φf (Γ
(p)0 , ρt)
]= E
[Φf (Γ
(p)t , ρ0)
]for all t ≥ 0.
![Page 68: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/68.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Sample interpretation of the duality
E
[Φf (Γ
(p)0 , ρt)
]= E
[Φf (Γ
(p)t , ρ0)
]ge
nerat
ions
20 40 60 80 100
20
40
60
80
100
![Page 69: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/69.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Galton-Watson processes and Cannings’ modelsContinuous state branching processesΛ-Fleming-Viot processesDuality
Pathwise duality via Donnelly & Kurtz’ (modified)
lookdown construction
new particleat level 3
new particleat level 6
post−birth types
pre−birth types
a
b
a
g
b
c
e
f
d
g
b
c
d
b
e
f
7
6
5
4
3
2
1
pre−birth labels
post−birth labels
9
8
7
6
5
4
3
2
1
![Page 70: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/70.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
TheoremHeuristics (discont. case)
Let (Xt) be a Dawson-Watanabe process (for simplicity, type space[0, 1], no mutation), Zt := Xt([0, 1]) its total mass process,Rt := Xt/Zt .
Theorem (B., Blath, Capaldo, Etheridge, Mohle, Schweinsberg,Wakolbinger (2005), Hiraba (2000), Perkins (1991))The ratio process (Rt)0≤t<τ can be time-changed with an additivefunctional of the total mass process (Zt) to obtain a Markovprocess if and only if
Z is a CSBP of some index α ∈ (0, 2].
If α = 2, Tt =∫ t
0 σ2Z−1
s ds and T−1(t) = inf{s : Ts > t}. Theprocess (RT−1(t))t≥0 is the classical (non-spatial) Fleming-Viotprocess, dual to Kingman’s coalescent.If α ∈ (0, 2), Tt = const ·
∫ t
0 Z 1−αs ds and (RT−1(t))t≥0 is the
Beta(2 − α,α)-Fleming-Viot process.
![Page 71: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/71.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
TheoremHeuristics (discont. case)
Let X and X ′ be two independent CSBP’s with the samecharacteristics (ν), St := Xt + X ′
t , Rt := Xt/St .
At rate St−ν(dh)dt, a new family of size h > 0 is created , therelative mass of the newborns is y := h/(St− + h), so
∆Rt =
{y(1 − R) with probability Rt− and−yR with probability (1 − Rt−) .
To eliminate the dependence of the relative jump size y on thecurrent total population size St−, ν must satisfy
sν({h : h/(h + s) > y}) = sν({h : h > sy/(1 − y)}) = g(s)f (y)
for all s, y > 0 for some functions f , g .
![Page 72: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/72.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
TheoremHeuristics (discont. case)
‘Meta-mathematic’ associations
Brownian motion ↔ Kingman’s coalescent∩ ∩
Stable processes ↔ Beta(2 − α,α)-coalescents∩ ∩
General Levy processes ↔ General Λ-coalescents
![Page 73: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/73.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Playing god with simulated “full trees”
1.0
1.0
true α
αM
L,fulltree
1.2
1.2
1.4
1.4
1.6
1.6
1.8
1.8
2.0
2.0
ML estimates of α for simulted datasets with sample size n = 100,estimate based on full genealogical tree
(400 replicates for each value of α).
![Page 74: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/74.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Consider n-Λ-coalescent with mutation rate r per line (and infinitealleles mutation model). n = (n1, . . . , nℓ), possible type
configuration
Theorem (Mohle) The probability p(n) of observing a typeconfiguration n = (n1, . . . , nℓ) satisfies the recursion given byp(1) = 1 and
p(n) =nr∑n
k=2
(nk
)λn,k + nr
ℓ∑
j=1nj =1
1
ℓp(n(j))
+1∑n
k=2
(nk
)λn,k + nr
n∑
k=2
ℓ∑
j=1nj≥k
(n
k
)λn,k
nj − k + 1
n − k + 1p(n − (k − 1)ej ).
(n(j) = (n1, . . . , nj−1, nj+1, . . . , nk) )
![Page 75: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/75.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Watterson’s infinitely many sites model
Model genetic locus as infinite sequence of completely linked sites,mutations always hit a new site.
Mathematical abstraction:
a gene is [0, 1]
a type is a configuration of points on [0, 1]
Ethier & Griffiths (1987) parametrisation:
type space E = [0, 1]N
mutation operator
Bf((x1, x2, . . . )
)= r
∫ 1
0f((u, x1, x2, . . . )
)−f
((x1, x2, . . . )
)du
![Page 76: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/76.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Asymptotics of the frequency spectrum
Consider an n-Beta(2 − α,α)-coalescent, mutations at rate r
according to the infinitely-many-sites model (assuming knownancestral types). Let
M(n) := #total number of mutations in the sample,
Mk(n) := #number of mutations affecting exaktly k samples,
k = 1, 2, . . . , n − 1.Theorem (Berestycki, Berestycki & Schweinsberg)
M(n)
n2−α→ r
α(α− 1)Γ(α)
2 − α,
Mk(n)
n2−α→ rα(α− 1)2
Γ(k + α− 2)
k!
in probability as n → ∞.
![Page 77: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/77.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Asymptotics of the frequency spectrum
Consider an n-Beta(2 − α,α)-coalescent, mutations at rate r
according to the infinitely-many-sites model (assuming knownancestral types). Let
M(n) := #total number of mutations in the sample,
Mk(n) := #number of mutations affecting exaktly k samples,
k = 1, 2, . . . , n − 1.Theorem (Berestycki, Berestycki & Schweinsberg)
M(n)
n2−α→ r
α(α− 1)Γ(α)
2 − α,
Mk(n)
n2−α→ rα(α− 1)2
Γ(k + α− 2)
k!
in probability as n → ∞.
Thus M1(n)/M(n) ≈ 2 − α for n large, which suggests
αBBS := 2 −M1(n)
M(n)as an estimator for α.
![Page 78: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/78.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Infinitely-many-sites model
Model genetic locus as infinite sequence of completely linked sites,mutations always hit a new site
Example: segr. siteSeq. 1 2 3 41 1 0 0 02 1 1 0 03 0 0 1 14 0 0 1 15 0 0 1 0
(0=wild type, 1=mutant
assume known ancestral types)
Obs. fit IMS ⇐⇒ no sub-matrix1 01 10 1
aa bb cc(nor row permutation).
![Page 79: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/79.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Infinitely-many-sites model, II
If the infinitely-many-sites model applies, the observationscorrespond to a unique rooted perfect phylogeny (or ‘genetree’).
Sequences, Genetree, obs. types
segr. siteSeq. 1 2 3 41 1 0 0 02 1 1 0 03 0 0 1 14 0 0 1 15 0 0 1 0
1
2
3
4
1 2 3,4 5
type multiplicity(1, 0) 1(2, 1, 0) 1(4, 3, 0) 2(3, 0) 1
Construct e.g. using Gusfield’s (1991) algorithm.
Note: purely combinatorial, does not depend on a probabilisticmodel for the observations.
![Page 80: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/80.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Simulating samples under the IMS model
The Ethier-Griffiths urn (1987) can be used to generate arandom sample of size n under Kingman’s coalescent(with mutation rate r per line):
Start with 2 leaves.
When there are k leaves:
Add a mutation to a leaf w. prob. 2r2r+(k−1) ,
split one leaf w. prob. k−12r+(k−1)
(leaf picked uniformly among the k).
Stop when n + 1 leaves, delete last leaf.
![Page 81: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/81.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Simulating samples under the IMS model: Λ-case
(Y(n)t )≥0 block counting process of Λ-coalescent starting from n
blocks:
Jump from i to j ∈ {1, 2, . . . , i − 1} at rateqij :=
(i
i−j+1
)λi ,i−j+1.
τ1 := inf{t ≥ 0 : Y(n)t = 1}.
Y(n)t := Y
(n)(τ1−t)− time-reversed block counting process
(Y(n)t = ∂ for t ≥ τ1).
Jump rates q(n)ji = gniqi j
gnj, q
(n)n∂ = −qnn =
∑n−1j=1 qnj ,
P(Y(n)0 = k) = P(Y
(n)τ1−
= k) = gnkqk1.
gni := E∫ ∞0 1(Y
(n)t = i) dt is the Green function (in general,
not known explicitly, but easy recursion).
![Page 82: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/82.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
A computer experimentInfinitely many allelesInfinitely many sites(Λ-) Ethier-Griffiths’ urn
Simulating samples under the IMS model: Λ-case, cont.
The n-Λ-“Ethier-Griffiths urn” (mutation rate r).
Begin with K leaves, P(K = k) = P(Y(n)0 = k).
While there are k leaves:
Add a mutation to a leaf w. prob. r
kr−eq(n)kk
,
split one leaf into ℓ w. prob.eq(n)
k,k+ℓ−1
eq(n)kk
,
if k = n goto stop w. prob. −eq(n)nn
kr−eq(n)nn
(leaf picked uniformly among the k).
Stop.
![Page 83: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/83.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Recursion for tree probabilities
Can calculate P(r ,Λ)
(observed sequence data (t,n)
)=: p(t,n) via
1 2
34
0
![Page 84: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/84.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Recursion for tree probabilities
Can calculate P(r ,Λ)
(observed sequence data (t,n)
)=: p(t,n) via
1 2
34
0
p(t,n) =1
rn + λn
∑
i :ni≥2
ni∑
k=2
(n
k
)λn,k
ni − k + 1
n − k + 1p(t,n− (k − 1)ei
)
+r
rn + λn
∑
i :ni =1,xi0unique,
s(xi )6=xj∀j
p(si (t),n
)
+r
rn + λn
1
d
∑
i :ni =1,
xi0unique
∑
j:s(xi )=xj
(nj + 1)p(ri (t), ri (n + ej)
).
Extends Ethier & Griffiths (1987) to Λ-coalescents andMohle’s recursion (2005) to IMS model.
![Page 85: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/85.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Compute probabilities
Use exact recursions for moderate sample complexities.Approach more complex samples by version of Griffiths &Tavare’s (1994)
Monte Carlo method
p(t,n) = E(t,n)
[ τ−1∏
i=0
f(r ,Λ)(Xi)]
For suitable Markov chain Xi on sample configurations.
Estimate expectation via empirical mean of independent runs.
extension to Λ-coalescents by B. & Blath (2008)
![Page 86: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/86.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Artifical sample of size 12 analysed with r = 1 and α = 1.5:
5.0 5.5 6.0 6.5 7.0
−2.
5−
2.0
−1.
5−
1.0
−0.
50.
0
Griffiths & Tavaré
log10(
b sdest
imate
)
log10(#runs)
![Page 87: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/87.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
HIStory
Interpret genealogy as sequency of historical states:H =
(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)
)
1
2
3
0
(((3, 1, 0), (1, 0), (2, 1, 0), (0)
), (1, 2, 1, 1)
)
![Page 88: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/88.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
HIStory
Interpret genealogy as sequency of historical states:H =
(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)
)
1
2
3
0
H0
H−1
H−2
H−3
H−4
H−5
H−6
(((3, 1, 0), (1, 0), (2, 1, 0), (0)
), (1, 2, 1, 1)
)
![Page 89: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/89.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
HIStory
Interpret genealogy as sequency of historical states:H =
(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)
)
1
2
3
0
1
2
3
0
(((3, 1, 0), (1, 0), (2, 1, 0), (0)
), (1, 2, 1, 1)
)
![Page 90: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/90.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
HIStory
Interpret genealogy as sequency of historical states:H =
(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)
)
1
2
3
0
1
2
3
0
1
2
3
0
(((3, 1, 0), (1, 0), (2, 1, 0), (0)
), (1, 2, 1, 1)
)
![Page 91: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/91.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
HIStory
Interpret genealogy as sequency of historical states:H =
(H−τ = ((1), (0)),Hτ−1 , . . . ,H−1,H0 = (t,n)
)
1
2
3
0
1
2
3
0
1
2
3
0
different histories can lead to same sample(((3, 1, 0), (1, 0), (2, 1, 0), (0)
), (1, 2, 1, 1)
)
![Page 92: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/92.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Importance sampling
We have
p(t,n) = P(r ,Λ)
(H0 = (t,n)
)=
∑
H:H0=(t,n)
P(r ,Λ)
(H
)
=∑
H:H0=(t,n)
P(r ,Λ)
(H
)
Q(H
)︸ ︷︷ ︸
=:w(H)importance weight
Q(H),
for any law Q on histories s.th. P(r ,Λ)
∣∣∣{H0=(t,n)}
≪ Q.
Thus,
p(t,n) ≈1
R
R∑
i=1
w(H(i)),
where H(1), . . . ,H(R) are independent samples from Q
![Page 93: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/93.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Simply the best
p(t,n) =∑
H:H0=(t,n)
P(r ,Λ)
(H
)
Q(H
) Q(H
)≈
1
R
R∑
i=1
w(H(i))
Qopt(·) := P(r ,Λ)
(·∣∣H0 = (t,n)
)is optimal (Stephens &
Donnelly 2000):
Variance of estimator is zero since w(H(i)) ≡ p(t,n).
Finding Qopt is as hard as the original problem.
H0,H−1, . . . is Markov chain under Qopt.
Remark: Transistion probabilities qGT(Hi |Hi+1) ∝ P(r ,Λ)(Hi+1|Hi )gives (Λ-)Griffiths-Tavare method.
![Page 94: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/94.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Stephens and Donnelly’s (2000) IMS candidate
Kingman case: Choose individual uniformly:
If type is unique in sample, remove “outmost” mutation,
if at least two individuals with this type, merge two lines.
(optimal proposal distribution for infinitely-many-alleles model,HUW (2008))
Heuristic extension to Λ case:
(t,n) →
(si (t),n
)w.p. ∝ 1 if ni = 1, xi0 unique, si (xi ) 6= xj∀j(
ri (t, ri (n + ej )) w.p. ∝ 1 if ni = 1, xi0 unique, si (xi ) 6= xj(t,n − (k − 1)ei
)w.p. ∝ ni qni
(k) if 2 ≤ k ≤ ni ,
where qni(k) =
qn,n−k+1Pnil=2 qn,n−l+1
, jump probabilities of block countingprocess.
![Page 95: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/95.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Artifical sample of size 12 analysed with r = 1 and α = 1.5:
5.0 5.5 6.0 6.5 7.0
−2.
5−
2.0
−1.
5−
1.0
−0.
50.
0
Griffiths & TavaréStephens & Donnelly
log10(
b sdest
imate
)
log10(#runs)
![Page 96: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/96.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea
Sample of size n where exactly one mutation is visible (in d copies).
0 n-d
1 d
p(1)(r ,Λ)(n, d) = P(r ,Λ)
{most recent event involves in-dividual bearing mutation
}
Probability can be computed
Kingman case: explicit formula (HUW (2008))
Λ case: numerically, using recursion
![Page 97: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/97.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
For a general sample (t,n)0 0
1 2
4 2 5 4
2 1 3 3
iput
ui ,m =
{
where mutation m is present in dm individuals.
![Page 98: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/98.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
For a general sample (t,n)0 0
1 2
4 2 5 4
2 1 3 3
i
carrying:
0 4
1 = d11 8
put
ui ,m =
{p
(1)(r ,Λ)(n, dm) · ni
dmif i bears m
where mutation m is present in dm individuals.
![Page 99: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/99.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
For a general sample (t,n)0 0
1 2
4 2 5 4
2 1 3 3
i
carrying:
0 4
1 8
0 8
5 4
put
ui ,m =
{p
(1)(r ,Λ)(n, dm) · ni
dmif i bears m
where mutation m is present in dm individuals.
![Page 100: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/100.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
For a general sample (t,n)0 0
1 2
4 2 5 4
2 1 3 3
i
carrying:
0 4
1 8
0 8
5 4
not carrying:
0 11
2 1
put
ui ,m =
{p
(1)(r ,Λ)(n, dm) · ni
dmif i bears m
(1 − p
(1)(r ,Λ)(n, dm)
)· ni
n−dmotherwise,
where mutation m is present in dm individuals.
![Page 101: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/101.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
For a general sample (t,n)0 0
1 2
4 2 5 4
2 1 3 3
i
carrying:
0 4
1 8
0 8
5 4
not carrying:
0 11
2 1
0 9
3 3
0 10
4 2
put
ui ,m =
{p
(1)(r ,Λ)(n, dm) · ni
dmif i bears m
(1 − p
(1)(r ,Λ)(n, dm)
)· ni
n−dmotherwise,
where mutation m is present in dm individuals.
![Page 102: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/102.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
For a general sample (t,n)0 0
1 2
4 2 5 4
2 1 3 3
i
carrying:
0 4
1 8
0 8
5 4
not carrying:
0 11
2 1
0 9
3 3
0 10
4 2
put
ui ,m =
{p
(1)(r ,Λ)(n, dm) · ni
dmif i bears m
(1 − p
(1)(r ,Λ)(n, dm)
)· ni
n−dmotherwise,
where mutation m is present in dm individuals. Propose type i
according to
qΛ-HUW
(i |(t,n)
)∝
{∑m ui ,m if i is allowed to act
0 otherwise.
![Page 103: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/103.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Hobolth, Uyenoyama & Wiuf’s (2008) idea contd.
If proposed type i
is singleton: remove “outmost” mutation,
has ni ≥ 2: merger inside type i .
Kingman case: merge two lines
Λ-case: propose ℓ+ 1-merger w.p. ∝ Pr,Λ
{0 n
1 do − l
∣∣∣∣0 n
1 do
}
![Page 104: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/104.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Artifical sample of size 12 analysed with r = 1 and α = 1.5:
5.0 5.5 6.0 6.5 7.0
−2.
5−
2.0
−1.
5−
1.0
−0.
50.
0
Griffiths & TavaréStephens & DonnellyHobolth, Uyenoyama & Wiuf
log10(
b sdest
imate
)
log10(#runs)
![Page 105: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/105.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
Tree probabilitiesImportance samplingPerformance
Performance
Simulated 50 samples of size 15 with r = 2 and α = 1.5. Analysedwith r = 1 and α = 1.5. Time needed to get relative error below0.01:
05
1015
20
3.9 4.5 5.1 5.7 6.3 6.9 7.5 8.1 8.7
(a) measured in log10(# runs of MC)
05
1015
-0.7 -0.2 0.3 0.8 1.3 1.8 2.3 2.8 3.3
(b) measured in log10(seconds)
![Page 106: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/106.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
IllustrationOutlook
Dataset from Ward et al, Extensive Mitochondrial
Diversity Within a Single Amerindian Tribe, PNAS 1991
Analysis with Beta-Coalescent:
1
2
3
4
5
1.2 1.4 1.6 1.8 2.0
1e−2
5
1e−25
1e−2
4
1e−241e−23
1e−2
3
1e−22
1e−2
2
1e−211e−21
1e−202e−20
4e−206e−20
8e−209e−20
α
r
Mitochondrial control region from 55 female Nuu-Chah-Nulth,αML = 1.9, rML = 2.2(Sample as edited in Griffiths & Tavare, Stat. Sci., 1994)
![Page 107: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/107.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
IllustrationOutlook
Genetic variation at the mitochondrial cyt b locus of Atlantic cod
Carr & Marshall 1991 Pepin & Carr 1993 Arnason & Palsson 1996
0.5
1.0
1.5
2.0
2.5
3.0
1.2 1.4 1.6 1.8 2.0
−15.0
−12.
0
−12.0−11.0
−11.0
−10.0
−10.0
−9.0
−8.7 −8.5
−15
−14
−13
−12
−11
−10
−9
−8
α
r
0.5
1.0
1.5
2.0
2.5
3.0
1.2 1.4 1.6 1.8 2.0
−15.0
−13.0
−13.0−12.0
−12.0
−11.5
−11.5
−11.0 −10.5
−10.0 −9.8
−20
−18
−16
−14
−12
−10
α
r0.5
1.0
1.5
2.0
2.5
3.0
1.2 1.4 1.6 1.8 2.0
−20.0
−18.0
−15.0
−15.0−14.0−14.0
−13.5
−13.0−12.7
−22
−21
−20
−19
−18
−17
−16
−15
−14
−13
α
r
bαML = 1.4,brML = 0.8 bαML = 1.4,brML = 0.6 bαML = 1.65,brML = 0.7
Arnason et al 1998 Arnason et al 2000 Sigurgıslason & Arnason 2003
0.5
1.0
1.5
2.0
2.5
3.0
1.2 1.4 1.6 1.8 2.0
−18.0
−18.0−15.0
−15.0
−14.5
−14.5
−14.0−13.7
−22
−21
−20
−19
−18
−17
−16
−15
−14
α
r
0.5
1.0
1.5
2.0
2.5
3.0
1.2 1.4 1.6 1.8 2.0
−15.0
−12.
0
−12.0−11.0
−11.0
−10.0
−10.0
−9.0
−8.7 −8.5
−20
−18
−16
−14
−12
−10
−8
α
r
0.5
1.0
1.5
2.0
2.5
3.0
1.2 1.4 1.6 1.8 2.0
−15.0
−12.
0
−12.0−11.0
−11.0
−10.0
−10.0
−9.0
−8.8
−8.5
−20
−18
−16
−14
−12
−10
−8
α
r
bαML = 1.55,brML = 0.6 bαML = 1.4,brML = 0.8 bαML = 1.4,brML = 0.8
![Page 108: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/108.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
IllustrationOutlook
Summary & Outlook
Eldon & Wakeley, Genetics 2006, wrote
For many species, the coalescent with multiple mergers might
be a better null model than Kingman’s coalescent.
![Page 109: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/109.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
IllustrationOutlook
Summary & Outlook
Eldon & Wakeley, Genetics 2006, wrote
For many species, the coalescent with multiple mergers might
be a better null model than Kingman’s coalescent.
For panmictic fixed-size discrete generations populations,haploid neutral one-locus theory is “mathematicallycomplete”.
Tools for estimation exist, results point towards“non-Kingman-ness” in certain cases.
Statistical properties of estimators?
speed-up of computer-intensive methods?combinations between IS-methods possibleDouble-HUW: ask all pairs of mutations what to do
A good class of alternative models?
![Page 110: Lambda-coalescents and population genetic inference...Neutral mutation models Computing likelihoods Illustration and outlook Lambda-coalescents and population genetic inference Matthias](https://reader033.vdocument.in/reader033/viewer/2022050305/5f6d5f3ba8351c1ec27b3345/html5/thumbnails/110.jpg)
IntroductionCSBPes and Fleming-Viot processes
Time change relationNeutral mutation models
Computing likelihoodsIllustration and outlook
IllustrationOutlook
Thank you for your attention!