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Estimating the divergence time ofprimates
An introduction to ABC lecture
June 11 2019
1
Statistical inference on trees: timescales
• Introduction
• Primate fossil record
• Dating splits by ABC
• Today’s posterior is tomorrow’s prior: molecular data
• Conclusions
2
Charles Darwin (1809 - 1882)
3
Ernst Haeckel (1834-1919)
4
Haeckel tree of life (detail)
5
Haeckel tree of life (detail)
6
August Schleicher (1821-1868)
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The Primates
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Primate Evolution
Time
t
9
Reconciling molecular and fossil records?
• Extant primates are strepsirrhines (lemurs and lorises)and haplorhines (tarsiers and anthropoids)
• Molecular estimate of time of divergence isapproximately 90 mya
• Fossil record suggests 60-65 mya
• Fossil record is patchy
Problem: Use the fossil record to estimate the age of thelast common ancestor of extant primates
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Primate Data
Tables
Table 1. Data and relative sampling intensities for the primate fossil record, taking a totalof 235 modern species. References for the data can be found in the Supplementary
Information.
Observed Relative RelativeEpoch k Tk number of sampling sampling
species (Dk) intensity (pk) intensity (pk)Scheme 1 Scheme 2
Late Pleistocene 1 0.15 19 1.0 1.0Middle Pleistocene 2 0.9 28 1.0 1.0Early Pleistocene 3 1.8 22 1.0 1.0Late Pliocene 4 3.6 47 1.0 1.0Early Pliocene 5 5.3 11 1.0 0.5Late Miocene 6 11.2 38 1.0 0.5Middle Miocene 7 16.4 46 1.0 1.0Early Miocene 8 23.8 36 1.0 0.5Late Oligocene 9 28.5 4 1.0 0.1Early Oligocene 10 33.7 20 1.0 0.5Late Eocene 11 37.0 32 1.0 1.0Middle Eocene 12 49.0 103 1.0 1.0Early Eocene 13 54.8 68 1.0 1.0Pre-Eocene 14 0 0.1 0.1
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The evolutionary process
t
B
A
T_k T_5 T_4 T_3 T_2 T_1
Time
12
What happened?
• Average sampling fraction of 5.7%
– upper 95% limit 7.4%
• Estimated divergence time 81.5 mya
– 95% CI (72.0, 89.6) mya
Tavare, Marshall, Will, Soligo & Martin Nature, 2002
• Pravda, Times, BBC, . . . , assorted religious fanatics,. . .
13
Primate Evolution
14
Why more?
• Bayesian approach more natural
• Allows us to incorporate prior information
• Sampling fractions
– probability of finding a fossil in bin i is αi
– ααα = αppp, ppp known
– reasonable?
• Other models for finds?
• Allowing for dinosaur extinction at K/T boundary?
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Fossil record: ABC approach
Data can be thought of in two parts:
(a) the observed number of fossils Fobs found
(b) the proportions pj,obs found in jth bin
A suitable metric might be∣∣∣∣ FFobs− 1
∣∣∣∣+
k+1∑j=1
|pj − pj,obs|
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Results ε = 0.1
0 500 1000 1500 2000
0.00
00.
002
0.00
4
Accepted N_0 values
dens
ity
0 20 40 60 80 100
0.00
0.02
0.04
Accepted tau values
dens
ity
0.00 0.05 0.10 0.15 0.20 0.25 0.30
01
23
45
6
Accepted alpha values
dens
ity
0.0 0.1 0.2 0.3 0.4 0.5
01
23
4
Accepted rho values
dens
ity
0.006 0.008 0.010 0.012 0.014
020
6010
0
Accepted gamma values
dens
ity
2.0 2.2 2.4 2.6 2.8 3.0
0.0
0.4
0.8
1.2
Accepted 1/lambda values
dens
ity
17
Some ABC technicalities
Hybrid ABC schemes
18
Sensitivity: Exploring Other Models
One advantage of ABC – it is easy to change the input . . .
- Choice of ρ
- Demography
- Sampling fractions
- K/T crash 65 mya
- the time of origin of primates is even furtherback in the Cretaceous
- Poisson sampling scheme: length in bin matters
- Dating other split points
19
Hybrid ABC schemes: ABC-Gibbs
J1 If currently at θθθ = (θ1, θ2), draw θ′1 from π(θ1|D, θ2)and set θθθ = (θ′1, θ2).
J2 Draw θ′2 from π(θ2) and simulate data D′ usingparameter θθθ = (θ′1, θ
′2).
J3 If D = D′, set θθθ = (θ′1, θ′2) and return to step J1.
Otherwise stay at θθθ = (θ′1, θ2) and return to step J2.
Steps J2 and J3 above are the mechanical version of therejection algorithm which gives samples from π(θ2|D, θ1).
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By replacing step J3 with
J3′ If ρ(D,D′) ≤ ε, set θθθ = (θ′1, θ′2) and return to step
J1. Otherwise stay at θθθ = (θ′1, θ2) and return to stepJ2.
we can generate approximate draws from π(θ2|D, θ1).
• Could also use Approximate Metropolis-within-Gibbsand other variants
21
Dealing with Sampling Fractions
f(λ, τ,N ,α|D) ∝ P(D|α,λ, τ,N )P(N|τ,λ)f(τ)f(λ)f(α)
where
• λ = (λ, γ, ρ) growth parameters,
• α = (α1, . . . , α14) sampling fractions
• N is the underlying tree structure
Give sampling fractions independent Beta(a, b) priors
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Gibbs-ABC Example
Split the random variable into two parts:α and (λ, τ,N )
Sample from the two conditional distributions
• f(α | D,λ, τ,N )
• f(τ,λ,N | D,α)
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Conditional distribution of α
f(α | D,λ, τ,N )
∝ f(α,λ, τ,N | D)
∝ P(N | τ, λ)f(τ)f(λ)f(α)P(D | τ,λ,N ,α)
∝ f(α)P(D | N ,α)
∝ Π14i=1α
dii (1− αi)Ni−diαa−1i (1− αi)b−1
∝ Πfβ(αi ; di + a,Ni − di + b)
Posterior mean of αi = a+diNi+a+b
≈ diNi
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Conditional distribution of (τ,λ,N )
f(τ,λ,N|D,α) ∝ f(λ, τ,N ,α|D)
∝ P(D|λ,α,N , α)P(N|τ, λ)f(τ)f(λ)
Simulate from this using ABC: accept (λ, τ,N ) ifρ(D,D′) < ε, where D′ represents the simulated data
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Metric and Priors
τ ∼ U [0, 100]
α ∼ U [0, 0.6]
ρ ∼ U [0, 0.8]
γ ∼ U [0.005, 0.015]
1/λ ∼ U [2, 3]
a = 0.1
b = 1
ε = 0.2
Same metric as before
26
No free lunches
0 200 400 600 800 1000
5010
015
020
025
030
0
Run number
N_0
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Tweak metric
• The observed N0 values are too small
– require N0 > 235
– change the metric
ρ(D,D′) =
k∑i=1
∣∣∣∣ Di
D+− D′iD′+
∣∣∣∣+
∣∣∣∣D′+D+− 1
∣∣∣∣+
∣∣∣∣N ′0N0− 1
∣∣∣∣• Penalises trees with N0 values far from 235
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Results: ε = 0.3
min LQ Median mean UQ Max
N0 184 212 224 226 238 279τ 0.0 8.0 18.6 26.3 36.8 99.5
180 200 220 240 260 280
0.0
00
0.0
10
0.0
20
Accepted N_0 values
density
0 20 40 60 80 100
0.0
00
0.0
15
0.0
30
Accepted tau values
density
0.0 0.2 0.4 0.6 0.8
01
23
45
Accepted rho values
density
0.006 0.008 0.010 0.012 0.014
040
80
120
Accepted gamma values, Strep
density
2.0 2.2 2.4 2.6 2.8 3.0
0.0
0.5
1.0
1.5
Accepted 1/lambda values.
density
29
Sensitivity: Exploring Other Models
One advantage of ABC – it is easy to change the input . . .
- Choice of ρ
- Demography
- Sampling fractions
- K/T crash 65 mya
- the time of origin of primates is even furtherback in the Cretaceous
- Poisson sampling scheme: length in bin matters
- Dating other split points
30
Old World/New World Split
Hap/Strep Plat/CatEpoch k Tk number of number of
species (Dk) species (D∗k)
Late Pleistocene 1 0.15 19 19Middle Pleistocene 2 0.9 28 28Early Pleistocene 3 1.8 22 22Late Pliocene 4 3.6 47 44Early Pliocene 5 5.3 11 10Late Miocene 6 11.2 38 33Middle Miocene 7 16.4 46 43Early Miocene 8 23.8 36 30Late Oligocene 9 28.5 4 3Early Oligocene 10 33.7 20 6Late Eocene 11 37.0 32 2Middle Eocene 12 49.0 103 0Early Eocene 13 54.8 68Pre-Eocene 14 0
31
Dating Two Splits
B
2
1
A
T5 T4 T3 T2 T1
! !
!
Tk
1
32
Details
• N0 = 235 species for the Strep/Hap,
• ε = 0.4 for both metrics
min LQ Median mean UQ MaxN0 159 212 234 233 254 303τ 0.9 12.1 17.6 20.1 25.3 94.5τ∗ 1.6 14.5 18.2 19.6 23.5 82.9
The median posterior sampling fractions (×100)α1 α2 α4 α5 α6 α8 α9 α10 α11 α12 α13 α148 10 12 3 6 7 1 8 22 41 80 18 8 8 4 4 4 1 4 8 8 8 1
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Posteriors
150 200 250 300
0.0
00
0.0
10
Accepted N_0 values
de
nsity
0 20 40 60 80 100
0.0
00
.02
0.0
4
Accepted tau values
de
nsity
0 20 40 60 80
0.0
00
.03
0.0
6
Accepted taustar values
de
nsity
0.0 0.2 0.4 0.6 0.8
02
46
81
2
Accepted rho values
de
nsity
0.006 0.008 0.010 0.012 0.014
04
08
01
20
Accepted gamma values, Strep
de
nsity
2.0 2.2 2.4 2.6 2.8 3.0
0.0
0.4
0.8
1.2
Accepted 1/lambda values.
de
nsity34
Dating Two Splits, revisited
B
2
1
A
T5 T4 T3 T2 T1
! !
!
Tk
1
35
The structure of branching processes
• Our approach to inferring multiple split points isheuristic
• What other approaches might work?
• Consider conditioning the process on a split at a fixedtime
– leads to a size-biassed GW process
– For ABC, need to be able to simulate theprocess
– Can use rejection . . .
36
A(nother) fishbone process
Time
GW
GW
GW
GW
37
Which metric?
ρ(D, X) =
14∑i=1
∣∣∣∣ Di
D+− Xi
X+
∣∣∣∣+
∣∣∣∣X+
D+− 1
∣∣∣∣+
∣∣∣∣X0
N0− 1
∣∣∣∣ .Match up:
• Proportions of fossils observed in each bin
• Total number of fossils observed
• Number of extant species
38
What happened?
39
Combining fossil record with molecular data
Yesterday’s posterior is tomorrow’s prior . . .
• Estimate posterior for two primate divergence times
• Use as prior for dating nodes from molecular data(mcmctree)
• Data are updated from earlier analysis
40
The posteriors
41
The phylogeny of the species (Poissonmodel)
42
The molecular data
43
References
• ST, Marshall C, Will O, Soligo C & Martin R (2002)Using the fossil record to estimate the age of the lastcommon ancestor of extant primates. Nature, 416,726–729
• Wilkinson R & ST (2009) Estimating primatedivergence times by using conditionedbirth-and-death processes. Theoretical PopulationBiology, 75, 278–295
• Wilkinson R, Steiper M, Soligo C, Martin R, Yang Z& ST (2011) Dating primate divergences through anintegrated analysis of palaeontological and moleculardata. Systematic Biology, 60, 16–31.
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