bayesian divergence time estimation
TRANSCRIPT
B D TETracy Heath
Ecology, Evolution, & Organismal BiologyIowa State University
@trayc7http://phyloworks.org
2015 SSB WorkshopAnn Arbor, MI USA
OLecture & Demo: Bayesian Fundamentals• Bayes theorem, priors and posteriors, MCMC• Example: the birth-death process• RevBayes Demonstration
breakLecture: Bayesian Inference of Species Divergence Times• Relaxed clock models – accounting for variation insubstitution rates among lineages
• Tree priors and fossil calibration
breakBEAST v2 Tutorial — dating species divergences under thefossilized birth-death process
lunchCourse materials: http://phyloworks.org/resources/ssbws.html
P BFor each of the topics in the pre-course survey, most of youfeel that you “have some experience, but there’s much moreto learn”.
diversification models
relaxed−clock models
Bayesian inference
model−based phylo
probability theory
unfa
mili
ar
I’ve h
eard
of it
fam
iliar, b
ut m
ore
to le
arn
pre
tty
com
fort
abl
e
expert
(29 participants responded to the survey)
B I
Estimate the probability of a hypothesis (model) conditionalon observed data.
The probability represents the researcher’s degree of belief.
Bayes’ Theorem specifies the conditional probability of thehypothesis given the data.
B’ T
Bayesian Fundamentals
B’ T
Bayesian Fundamentals
B’ T
Bayesian Fundamentals
B’ T
Bayesian Fundamentals
B’ T
Bayesian Fundamentals
B’ T
The posterior probability of a discrete parameter δconditional on the data D is
Pr(δ | D) =Pr(D | δ)Pr(δ)∑δ Pr(D | δ)Pr(δ)
∑δ Pr(D | δ)Pr(δ) is the likelihood marginalized over allpossible values of δ.
Bayesian Fundamentals
B’ T
The posterior probability density a continuous parameter θconditional on the data D is
f(θ | D) =f(D | θ)f(θ)∫
θ f(D | θ)f(θ)dθ
∫θ f(D | θ)f(θ)dθ is the likelihood marginalized over allpossible values of θ.
Bayesian Fundamentals
P
The the distribution of θ before any data are collected isthe prior
f(θ)
The prior describes your uncertainty in the parameters ofyour model.
Bayesian Fundamentals
PWe may assume a gamma-prior distribution on θ with ashape parameter α and a scale parameter β.
θ ∼ Gamma(α, β)
f(θ | α, β) = 1Γ(α)βαθα−1e
− θβ
0 1 2 3
Density
θ
This requires us to assign values for α and β based on ourprior belief, or we can place hyperpriors on theseparameters if we are uncertain about their values.
Bayesian Fundamentals
E: T B-D PA time machine allowed us to observe the dates of eachspeciation event in the history of extant bears
010
20
30
Oligocene Miocene Plio. Plei.
Paleogene Neogene Quat.
33.24
27.9
15.9
13.1
11.6
9.95
2.8
Example
E: T B-D P
We assume that thediversification of bearsmatches a birth deathprocess with parameters:
λ = speciation rateμ = extinction rate
010
20
30
Oligocene Miocene Plio. Plei.
Paleogene Neogene Quat.
33.24
27.9
15.9
13.1
11.6
9.95
2.8
Example
E: T B-D P
The birth-death process allows us to compute the probabilitydensity of our observed time-tree (Ψ) conditional on anyvalue of speciation (λ) and any value of extinction (μ).
f(Ψ | λ, μ)
This model states that Ψ ∼ BD(λ, μ)
Example
E: T B-D P
Another way of expressing Ψ ∼ BD(λ, μ) is with aprobabilistic graphical model
Our “observed” time-tree isconditioned on someconstant value of λ and μ.
Example
E: T B-D PIf our time machine also allowed us to know the rate ofspeciation and extinction, we can easily calculate thelikelihood of our observed tree:
f(Ψ | λ, μ) = N!(λ−μ)λN−1 e−(λ−μ)x1
λ− μe−(λ−μ)x1
N−1∏i=1
(λ− μ)2e−(λ−μ)x1
(λ− μe−(λ−μ)x1)2
f(Ψ | λ = 0.5, μ = 0.1) = 3.49215e−32
Example
RB D: T B-D P
RevBayes
Fully integrative Bayesian inference ofphylogenetic parameters usingprobabilistic graphical models and aninterpreted language
http://RevBayes.com
Example
G M RB
Graphical models provide tools forvisually & computationally representingcomplex, parameter-rich probabilisticmodels
We can depict the conditionaldependence structure of variousparameters and other random variables
Höhna, Heath, Boussau, Landis, Ronquist, Huelsenbeck. 2014.Probabilistic Graphical Model Representation in Phylogenetics.Systematic Biology. (doi: 10.1093/sysbio/syu039)
RB D: T B-D PThe Rev language specifying a birth-death model on thebear phylogeny with fixed values for λ and μ.
Example
E: T B-D P
What if we do not know λ and μ?
We can use frequentist or Bayesian methods for estimatingtheir values.
Frequentist methods require us to find the values of λ andμ that maximize f(Ψ | λ, μ).
Bayesian methods use prior distributions to describe ouruncertainty in λ and μ and estimate f(λ, μ | Ψ).
Example
E: T B-D P
We must define prior distributions for λ and μ to estimatethe posterior probability density
f(λ, μ | Ψ, y, z) =f(Ψ | λ, μ)f(λ | y)f(μ | z)∫
λ∫μ f(Ψ | λ, μ)f(λ | y)f(μ | z)dλdμ
Now y and z are the parameters of the prior distributionson λ and μ.
Example
E: T B-D P
We can choose exponential prior distributions for λ and μ.Now y and z represent the rate parameters of theexponential priors.
λ ∼ Exponential(y)μ ∼ Exponential(z)
f(λ | y) = ye−yλ
f(μ | z) = ze−zμ
Example
RB D: T B-D PThe Rev language specifying a birth-death model on thebear phylogeny with exponential priors on λ and μ.
Example
E: T B-D PNow that we have a defined model, how do we estimatethe posterior probability density?
λ ∼ Exponential(y)μ ∼ Exponential(z)
f(λ, μ | Ψ, y, z) =f(Ψ | λ, μ)f(λ | y)f(μ | z)∫
λ∫μ f(Ψ | λ, μ)f(λ | y)f(μ | z)dλdμ
Example
M C M C (MCMC)
An algorithm for approximating the posterior distribution
Metropolis, Rosenblusth, Rosenbluth, Teller, Teller. 1953. Equations of state calculations by fast computingmachines. J. Chem. Phys.
Hastings. 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika.
Bayesian Fundamentals
M C M C (MCMC)
More on MCMC from Paul Lewis—our esteemed SSBPresident—and his lecture on Bayesian phylogenetics
Slides source: https://molevol.mbl.edu/index.php/Paul_Lewis
Bayesian Fundamentals
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 42
MCMC robot’s rules
Uphill steps are always accepted
Slightly downhill steps are usually accepted
Drastic “off the cliff” downhill steps are almost never accepted
With these rules, it is easy to see why the
robot tends to stay near the tops of hills
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 43
(Actual) MCMC robot rules
Uphill steps are always accepted because R > 1
Slightly downhill steps are usually accepted because R is near 1
Drastic “off the cliff” downhill steps are almost never accepted because R is near 0
Currently at 1.0 m Proposed at 2.3 m R = 2.3/1.0 = 2.3
Currently at 6.2 m Proposed at 5.7 m R = 5.7/6.2 =0.92 Currently at 6.2 m
Proposed at 0.2 m R = 0.2/6.2 = 0.03
6
8
4
2
0
10
The robot takes a step if it draws a Uniform(0,1) random deviate that is less than or equal to R
=
f(D|�⇤)f(�⇤)f(D)
f(D|�)f(�)f(D)
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 44
Cancellation of marginal likelihood
When calculating the ratio R of posterior densities, the marginal probability of the data cancels.
f(�⇤|D)
f(�|D)
Posterior odds
=f(D|�⇤)f(�⇤)f(D|�)f(�)
Likelihood ratio Prior odds
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 45
Target vs. Proposal Distributions
Pretend this proposal distribution allows good mixing. What does good
mixing mean?
default2.TXT
State0 2500 5000 7500 10000 12500 15000 17500
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 46
Trace plots
“White noise” appearance is a sign of good mixing
I used the program Tracer to create this plot: http://tree.bio.ed.ac.uk/software/tracer/ !
AWTY (Are We There Yet?) is useful for investigating convergence:
http://king2.scs.fsu.edu/CEBProjects/awty/awty_start.php
log(
post
erio
r)
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 47
Target vs. Proposal Distributions
Proposal distributions with smaller variance...
Disadvantage: robot takes smaller steps, more time required to explore the same area
Advantage: robot seldom refuses to take proposed steps
smallsteps.TXT
State0 2500 5000 7500 10000 12500 15000 17500
-6
-5
-4
-3
-2
-1
0
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 48
If step size is too small, large-scale trends will be apparentlo
g(po
ster
ior)
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 49
Target vs. Proposal Distributions
Proposal distributions with larger variance...
Disadvantage: robot often proposes a step that would take it off a cliff, and refuses to move
Advantage: robot can potentially cover a lot of ground quickly
bigsteps2.TX
T
State0 2500 5000 7500 10000 12500 15000 17500
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
Paul O. Lewis (2014 Woods Hole Molecular Evolution Workshop) 50
Chain is spending long periods of time “stuck” in one place
“Stuck” robot is indicative of step sizes that are too large (most proposed steps would take the robot “off the cliff”)
log(
post
erio
r)
M C M C (MCMC)Thanks, Paul!
Slides source: https://molevol.mbl.edu/index.php/Paul_Lewis
See MCMCRobot, a helpfulsoftware program for learningMCMC by Paul Lewis
http://www.mcmcrobot.org
Bayesian Fundamentals
RB D: T B-D PThe Rev language specifying the MCMC sampler for thebirth-death model.
Example
RB D: T B-D PThe trace-plot of the MCMC samples for speciation rate
spe
cia
tio
n
State0 5000 10000 15000 20000 25000 30000
0
2.5E-2
5E-2
7.5E-2
0.1
0.125
0.15
0.175
Example
RB D: T B-D PMarginal posterior densities of the speciation rate andextinction rate.
De
nsi
ty
Combined
extinction
speciation
0 5E-2 0.1 0.15 0.2 0.250
10
20
30
40
50
Example
E: T B-D PAlas, we do not have a time machine and we do not knowlineage divergence times without error.
010
20
30
Oligocene Miocene Plio. Plei.
Paleogene Neogene Quat.
33.24
27.9
15.9
13.1
11.6
9.95
2.8
Example
D T E
Goal: Estimate the ages of interior nodes to understand thetiming and rates of evolutionary processes
Model how rates aredistributed across the tree
Describe the distribution ofspeciation events over time
External calibrationinformation for estimates ofabsolute node times
fossil Ailuropodinae
fossil Arctodus
fossil Ursus
fossil pinnipeds
Gray wolf
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Urs
ida
e
Time (My)
60 2040 0
Eocene Oligocene Miocene
Plio
Ple
is
Paleocene
stem fossil Ursidae
fossil canids
(Figure from Heath et al., PNAS 2014)
OLecture & Demo: Bayesian Fundamentals• Bayes theorem, priors and posteriors, MCMC• Example: the birth-death process• RevBayes Demonstration
breakLecture: Bayesian Inference of Species Divergence Times• Relaxed clock models – accounting for variation insubstitution rates among lineages
• Tree priors and fossil calibration
breakBEAST v2 Tutorial — dating species divergences under thefossilized birth-death process
lunchCourse materials: http://phyloworks.org/resources/ssbws.html
OLecture & Demo: Bayesian Fundamentals• Bayes theorem, priors and posteriors, MCMC• Example: the birth-death process• RevBayes Demonstration
breakLecture: Bayesian Inference of Species Divergence Times• Relaxed clock models – accounting for variation insubstitution rates among lineages
• Tree priors and fossil calibration
breakBEAST v2 Tutorial — dating species divergences under thefossilized birth-death process
lunchCourse materials: http://phyloworks.org/resources/ssbws.html
D T E
Goal: Estimate the ages of interior nodes to understand thetiming and rates of evolutionary processes
Model how rates aredistributed across the tree
Describe the distribution ofspeciation events over time
External calibrationinformation for estimates ofabsolute node times
fossil Ailuropodinae
fossil Arctodus
fossil Ursus
fossil pinnipeds
Gray wolf
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Urs
ida
e
Time (My)
60 2040 0
Eocene Oligocene Miocene
Plio
Ple
is
Paleocene
stem fossil Ursidae
fossil canids
(Figure from Heath et al., PNAS 2014)
A T-S EPhylogenetic trees can provide both topological informationand temporal information
100 0.020.040.060.080.0
EquusRhinocerosBosHippopotamusBalaenopteraPhyseterUrsusCanisFelisHomoPanGorillaPongoMacacaCallithrixLorisGalagoDaubentoniaVareciaEulemurLemurHapalemurPropithecusLepilemur
MirzaM. murinusM. griseorufus
M. myoxinusM. berthaeM. rufus1M. tavaratraM. rufus2M. sambiranensisM. ravelobensis
Cheirogaleus
Sim
iiform
es
Mic
roce
bu
s
Cretaceous Paleogene Neogene Q
Time (Millions of years)
Understanding Evolutionary Processes (Yang & Yoder Syst. Biol. 2003; Heath et al. MBE 2012)
T G M C
Assume that the rate ofevolutionary change isconstant over time
(branch lengths equalpercent sequencedivergence) 10%
400 My
200 My
A B C
20%
10%10%
(Based on slides by Jeff Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
T G M C
We can date the tree if weknow the rate of change is1% divergence per 10 My N
A B C
20%
10%10%
10%200 My
400 My
200 My
(Based on slides by Jeff Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
T G M C
If we found a fossil of theMRCA of B and C, we canuse it to calculate the rateof change & date the rootof the tree
N
A B C
20%
10%10%
10%200 My
400 My
(Based on slides by Jeff Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
R G M CRates of evolution vary across lineages and over time
Mutation rate:Variation in• metabolic rate• generation time• DNA repair
Fixation rate:Variation in• strength and targets ofselection
• population sizes
10%
400 My
200 My
A B C
20%
10%10%
U A
Sequence data provideinformation about branchlengths
In units of the expected # ofsubstitutions per site
branch length = rate × time0.2 expected
substitutions/site
Ph
ylo
ge
ne
tic R
ela
tio
nsh
ips
Se
qu
en
ce
Da
ta
R T
The sequence dataprovide informationabout branch length
for any possible rate,there’s a time that fitsthe branch lengthperfectly
0
1
2
3
4
5
0 1 2 3 4 5
Bra
nch
Ra
te
Branch Time
time = 0.8rate = 0.625
branch length = 0.5
(based on Thorne & Kishino, 2005)
R TThe expected # of substitutions/site occurring along abranch is the product of the substitution rate and time
length = rate × time length = rate length = time
Methods for dating species divergences estimate thesubstitution rate and time separately
B D T E
length = rate length = time
R = (r, r, r, . . . , rN−)
A = (a, a, a, . . . , aN−)
N = number of tips
B D T E
length = rate length = time
R = (r, r, r, . . . , rN−)
A = (a, a, a, . . . , aN−)
N = number of tips
B D T E
Posterior probability
f (R,A, θR, θA, θs | D,Ψ)
R Vector of rates on branchesA Vector of internal node ages
θR, θA, θs Model parametersD Sequence dataΨ Tree topology
B D T E
f(R,A, θR, θA, θs | D) =
f (D |R,A, θs) f(R | θR) f(A | θA) f(θs)f(D)
f(D |R,A, θR, θA, θs) Likelihoodf(R | θR) Prior on rates
f(A | θA) Prior on node agesf(θs) Prior on substitution parametersf(D) Marginal probability of the data
B D T E
Estimating divergence times relies on 2 main elements:
• Branch-specific rates: f (R | θR)
• Node ages: f (A | θA,C)
M R VSome models describing lineage-specific substitution ratevariation:
• Global molecular clock (Zuckerkandl & Pauling, 1962)• Local molecular clocks (Hasegawa, Kishino & Yano 1989;Kishino & Hasegawa 1990; Yoder & Yang 2000; Yang & Yoder2003, Drummond and Suchard 2010)
• Punctuated rate change model (Huelsenbeck, Larget andSwofford 2000)
• Log-normally distributed autocorrelated rates (Thorne,Kishino & Painter 1998; Kishino, Thorne & Bruno 2001; Thorne &Kishino 2002)
• Uncorrelated/independent rates models (Drummond et al.2006; Rannala & Yang 2007; Lepage et al. 2007)
• Mixture models on branch rates (Heath, Holder, Huelsenbeck2012)
Models of Lineage-specific Rate Variation
G M C
The substitution rate isconstant over time
All lineages share the samerate
branch length = substitution rate
low high
Models of Lineage-specific Rate Variation (Zuckerkandl & Pauling, 1962)
R-C M
To accommodate variation in substitution rates‘relaxed-clock’ models estimate lineage-specific substitutionrates
• Local molecular clocks• Punctuated rate change model• Log-normally distributed autocorrelated rates• Uncorrelated/independent rates models• Mixture models on branch rates
L M C
Rate shifts occurinfrequently over the tree
Closely related lineageshave equivalent rates(clustered by sub-clades)
low high
branch length = substitution rate
Models of Lineage-specific Rate Variation (Yang & Yoder 2003, Drummond and Suchard 2010)
L M C
Most methods forestimating local clocksrequired specifying thenumber and locations ofrate changes a prioriDrummond and Suchard(2010) introduced aBayesian method thatsamples over a broad rangeof possible random localclocks
low high
branch length = substitution rate
Models of Lineage-specific Rate Variation (Yang & Yoder 2003, Drummond and Suchard 2010)
A R
Substitution rates evolvegradually over time –closely related lineages havesimilar rates
The rate at a node isdrawn from a lognormaldistribution with a meanequal to the parent rate
low high
branch length = substitution rate
Models of Lineage-specific Rate Variation (Thorne, Kishino & Painter 1998; Kishino, Thorne & Bruno 2001)
P R C
Rate changes occur alonglineages according to apoint process
At rate-change events, thenew rate is a product ofthe parent’s rate and aΓ-distributed multiplier
low high
branch length = substitution rate
Models of Lineage-specific Rate Variation (Huelsenbeck, Larget and Swofford 2000)
I/U R
Lineage-specific rates areuncorrelated when the rateassigned to each branch isindependently drawn froman underlying distribution
low high
branch length = substitution rate
Models of Lineage-specific Rate Variation (Drummond et al. 2006)
I M M
Dirichlet process prior:Branches are partitionedinto distinct rate categories
Random variables under theDPP informed by the data:• the number of rateclasses
• the assignment ofbranches to classes
• the rate value for eachclass
branch length = substitution rate
c5
c4
c3
c2
substitution rate classes
c1
Models of Lineage-specific Rate Variation (Heath, Holder, Huelsenbeck. 2012 MBE)
M R V
These are only a subset of the available models forbranch-rate variation
• Global molecular clock• Local molecular clocks• Punctuated rate change model• Log-normally distributed autocorrelated rates• Uncorrelated/independent rates models• Dirchlet process prior
Models of Lineage-specific Rate Variation
M R VAre our models appropriate across all data sets?
cave bear
American
black bear
sloth bear
Asian
black bear
brown bear
polar bear
American giant
short-faced bear
giant panda
sun bear
harbor seal
spectacled
bear
4.08
5.39
5.66
12.86
2.75
5.05
19.09
35.7
0.88
4.58
[3.11–5.27]
[4.26–7.34]
[9.77–16.58]
[3.9–6.48]
[0.66–1.17]
[4.2–6.86]
[2.1–3.57]
[14.38–24.79]
[3.51–5.89]14.32
[9.77–16.58]
95% CI
mean age (Ma)
t 2
t 3
t 4
t 6
t 7
t 5
t 8
t 9
t 10
t x
node
MP•MLu•MLp•Bayesian
100•100•100•1.00
100•100•100•1.00
85•93•93•1.00
76•94•97•1.00
99•97•94•1.00
100•100•100•1.00
100•100•100•1.00
100•100•100•1.00
t 1
Eocene Oligocene Miocene Plio Plei Hol
34 5.3 1.823.8 0.01
Epochs
Ma
Global expansion of C4 biomassMajor temperature drop and increasing seasonality
Faunal turnover
Krause et al., 2008. Mitochondrial genomes reveal anexplosive radiation of extinct and extant bears near theMiocene-Pliocene boundary. BMC Evol. Biol. 8.
Taxa
1
5
10
50
100
500
1000
5000
10000
20000
0100200300MYA
Ophidiiformes
Percomorpha
Beryciformes
Lampriformes
Zeiforms
Polymixiiformes
Percopsif. + Gadiif.
Aulopiformes
Myctophiformes
Argentiniformes
Stomiiformes
Osmeriformes
Galaxiiformes
Salmoniformes
Esociformes
Characiformes
Siluriformes
Gymnotiformes
Cypriniformes
Gonorynchiformes
Denticipidae
Clupeomorpha
Osteoglossomorpha
Elopomorpha
Holostei
Chondrostei
Polypteriformes
Clade r ε ΔAIC
1. 0.041 0.0017 25.32. 0.081 * 25.53. 0.067 0.37 45.1 4. 0 * 3.1Bg. 0.011 0.0011
Ostariophysi
Acanthomorpha
Teleo
stei
Santini et al., 2009. Did genome duplication drive the originof teleosts? A comparative study of diversification inray-finned fishes. BMC Evol. Biol. 9.
M R V
These are only a subset of the available models forbranch-rate variation
• Global molecular clock• Local molecular clocks• Punctuated rate change model• Log-normally distributed autocorrelated rates• Uncorrelated/independent rates models• Dirchlet process prior
Model selection and model uncertainty are very importantfor Bayesian divergence time analysis
Models of Lineage-specific Rate Variation
B D T E
Estimating divergence times relies on 2 main elements:
• Branch-specific rates: f (R | θR)
• Node ages: f (A | θA,C)
http://bayesiancook.blogspot.com/2013/12/two-sides-of-same-coin.html
P N T
Relaxed clock Bayesian analyses require a prior distributionon node times
f(A | θA)
Different node-age priors make different assumptions aboutthe timing of divergence events
Node Age Priors
G N T P
Assumed to be vague or uninformative by not makingassumptions about biological processes
Uniform prior: the time ata given node has equalprobability across theinterval between the timeof the parent node and thetime of the oldest daughternode(conditioned on root age)
Node Age Priors
S B P
Node-age priors based on stochastic models of lineagediversification
Yule process: assumes aconstant rate of speciation,across lineagesA pure birth process—everynode leaves extantdescendants (no extinction)
Node Age Priors
S B P
Node-age priors based on stochastic models of lineagediversification
Constant-rate birth-deathprocess: at any point intime a lineage can speciateat rate λ or go extinct witha rate of μ
Node Age Priors
S B P
Node-age priors based on stochastic models of lineagediversification
Constant-rate birth-deathprocess: at any point intime a lineage can speciateat rate λ or go extinct witha rate of μ
Node Age Priors
S B P
Node-age priors based on stochastic models of lineagediversification
Constant-rate birth-deathprocess: at any point intime a lineage can speciateat rate λ or go extinct witha rate of μ
Node Age Priors
S B P
Different values of λ and μ leadto different trees
Bayesian inference under thesemodels can be very sensitive tothe values of these parameters
Using hyperpriors on λ and μaccounts for uncertainty in thesehyperparameters
Node Age Priors
S B P
Node-age priors based on stochastic models of lineagediversificationBirth-death-samplingprocess: an extension ofthe constant-rate birth-deathmodel that accounts forrandom sampling of tipsConditions on a probabilityof sampling a tip, ρ
Node Age Priors
P N T
Sequence data are only informative on relative rates & timesNode-time priors cannot give precise estimates of absolutenode ages
We need external information (like fossils) to calibrate orscale the tree to absolute time
Node Age Priors
C D T
Fossils (or other data) are necessary to estimate absolutenode ages
There is no information inthe sequence data forabsolute timeUncertainty in theplacement of fossils
N
A B C
20%
10%10%
10%200 My
400 My
C D
Bayesian inference is well suited to accommodatinguncertainty in the age of the calibration node
Divergence times arecalibrated by placingparametric densities oninternal nodes offset by ageestimates from the fossilrecord
N
A B C
200 My
De
nsity
Age
A F CMisplaced fossils can affect node age estimates throughoutthe tree – if the fossil is older than its presumed MRCA
Calibrating the Tree (figure from Benton & Donoghue Mol. Biol. Evol. 2007)
A F C
Crown clade: allliving species andtheir most-recentcommon ancestor(MRCA)
Calibrating the Tree (figure from Benton & Donoghue Mol. Biol. Evol. 2007)
A F C
Stem lineages:purely fossil formsthat are closer totheir descendantcrown clade thanany other crownclade
Calibrating the Tree (figure from Benton & Donoghue Mol. Biol. Evol. 2007)
A F C
Fossiliferoushorizons: thesources in therock record forrelevant fossils
Calibrating the Tree (figure from Benton & Donoghue Mol. Biol. Evol. 2007)
F C
Age estimates from fossilscan provide minimum timeconstraints for internalnodes
Reliable maximum boundsare typically unavailable
Minimum age Time (My)
Calibrating Divergence Times
P D C N
Common practice in Bayesian divergence-time estimation:
Parametric distributions aretypically off-set by the ageof the oldest fossil assignedto a clade
These prior densities do not(necessarily) requirespecification of maximumbounds
Uniform (min, max)
Exponential (λ)
Gamma (α, β)
Log Normal (µ, σ2)
Time (My)Minimum age
Calibrating Divergence Times
P D C N
Calibration densities describethe waiting time betweenthe divergence event andthe age of the oldest fossil
Minimum age
Exponential (λ)
Time (My)
Calibrating Divergence Times
P D C N
Common practice in Bayesian divergence-time estimation:
Estimates of absolute nodeages are driven primarily bythe calibration density
Specifying appropriatedensities is a challenge formost molecular biologists
Uniform (min, max)
Exponential (λ)
Gamma (α, β)
Log Normal (µ, σ2)
Time (My)Minimum age
Calibration Density Approach
I F C
We would prefer toeliminate the need forad hoc calibrationprior densities
Calibration densitiesdo not account fordiversification of fossils
Domestic dog
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Zaragocyon daamsi
Ballusia elmensis
Ursavus brevihinus
Ailurarctos lufengensis
Ursavus primaevus
Agriarctos spp.
Kretzoiarctos beatrix
Indarctos vireti
Indarctos arctoides
Indarctos punjabiensis
Giant short-faced bear
Cave bear
Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C
We want to use allof the available fossils
Example: Bears12 fossils are reducedto 4 calibration ageswith calibration densitymethods
Domestic dog
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Zaragocyon daamsi
Ballusia elmensis
Ursavus brevihinus
Ailurarctos lufengensis
Ursavus primaevus
Agriarctos spp.
Kretzoiarctos beatrix
Indarctos vireti
Indarctos arctoides
Indarctos punjabiensis
Giant short-faced bear
Cave bear
Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C
We want to use allof the available fossils
Example: Bears12 fossils are reducedto 4 calibration ageswith calibration densitymethods
Domestic dog
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Zaragocyon daamsi
Ballusia elmensis
Ursavus brevihinus
Ailurarctos lufengensis
Ursavus primaevus
Agriarctos spp.
Kretzoiarctos beatrix
Indarctos vireti
Indarctos arctoides
Indarctos punjabiensis
Giant short-faced bear
Cave bear
Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C
Because fossils arepart of thediversification process,we can combine fossilcalibration withbirth-death models
Domestic dog
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Zaragocyon daamsi
Ballusia elmensis
Ursavus brevihinus
Ailurarctos lufengensis
Ursavus primaevus
Agriarctos spp.
Kretzoiarctos beatrix
Indarctos vireti
Indarctos arctoides
Indarctos punjabiensis
Giant short-faced bear
Cave bear
Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C
This relies on abranching model thataccounts forspeciation, extinction,and rates offossilization,preservation, andrecovery
Domestic dog
Spotted seal
Giant panda
Spectacled bear
Sun bear
Am. black bear
Asian black bear
Brown bear
Polar bear
Sloth bear
Zaragocyon daamsi
Ballusia elmensis
Ursavus brevihinus
Ailurarctos lufengensis
Ursavus primaevus
Agriarctos spp.
Kretzoiarctos beatrix
Indarctos vireti
Indarctos arctoides
Indarctos punjabiensis
Giant short-faced bear
Cave bear
Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
T F B-D P (FBD)
Improving statistical inference of absolute node ages
Eliminates the need to specify arbitrarycalibration densities
Better capture our statisticaluncertainty in species divergence dates
All reliable fossils associated with aclade are used
Useful for calibration or ‘total-evidence’dating
150 100 50 0
Time
(Heath, Huelsenbeck, Stadler. 2014 PNAS)
T F B-D P (FBD)
Recovered fossil specimensprovide historicalobservations of thediversification process thatgenerated the tree ofextant species
150 100 50 0
Time
Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B-D P (FBD)
The probability of the treeand fossil observationsunder a birth-death modelwith rate parameters:
λ = speciationμ = extinctionψ = fossilization/recovery
150 100 50 0
Time
Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B-D P (FBD)
The probability of the treeand fossil observationsunder a birth-death modelwith rate parameters:
λ = speciationμ = extinctionψ = fossilization/recovery
Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B-D P (FBD)
We use MCMC to samplerealizations of thediversification process,integrating over thetopology—includingplacement of thefossils—and speciation times
0250 50100150200
Time (My)
Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
I FBD TExtensions of the fossilized birth-death process accommodatevariation in fossil sampling, non-random species sampling, &shifts in diversification rates.
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Lower
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Paleocene
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Jurassic Cretaceous Paleogene Neogene Q.
With character data for both fossil & extant species, weaccount for uncertainty in fossil placement
S B-D P
A piecewise shifting modelwhere parameters changeover timeUsed to estimateepidemiological parametersof an outbreak
0175 255075100125150
Days
(see Stadler et al. PNAS 2013 and Stadler et al. PLoS Currents Outbreaks 2014)
S B-D Pl is the number ofparameter intervalsRi is the effectivereproductive numberfor interval i ∈ lδ is the rate ofbecomingnon-infectiouss is the probability ofsampling an individualafter becomingnon-infectious
S B-D P
l is the number ofparameter intervalsλi is the transmissionrate for interval i ∈ lμ is the viral lineagedeath rateψ is the rate eachindividual is sampled
S B-D P
S B-D P
A decline in R over thehistory of HIV-1 in the UKis consistent with theintroduction of effectivedrug therapies
After 1998 R decreasedbelow 1, indicating adeclining epidemic
(Stadler et al. PNAS 2013)
OLecture & Demo: Bayesian Fundamentals• Bayes theorem, priors and posteriors, MCMC• Example: the birth-death process• RevBayes Demonstration
breakLecture: Bayesian Inference of Species Divergence Times• Relaxed clock models – accounting for variation insubstitution rates among lineages
• Tree priors and fossil calibration
breakBEAST v2 Tutorial — dating species divergences under thefossilized birth-death process
lunchCourse materials: http://phyloworks.org/resources/ssbws.html
D T E S
Program Models/Methodr8s Strict clock, local clocks, NPRS, PLape (R) NPRS, PLmultidivtime log-n autocorrelated (plus some others)PhyBayes OU, log-n autocorrelated (plus some others)PhyloBayes CIR, white noise (uncorrelated) (plus some others)BEAST Uncorrelated (log-n & exp), local clocks (plus others)TreeTime Dirichlet model, CPP, uncorrelatedMrBayes 3.2 CPP, strict clock, autocorrelated, uncorrelatedDPPDiv DPP, strict clock, uncorrelatedRevBayes “the limit is the sky”∗
∗and other methods
E: C Y O ADating Bear DivergenceTimes with the FossilizedBirth-Death Process
Agriarctos spp X
Ailurarctos lufengensis X
Ailuropoda melanoleuca
Arctodus simus X
Ballusia elmensis X
Helarctos malayanus
Indarctos arctoides X
Indarctos punjabiensis X
Indarctos vireti X
Kretzoiarctos beatrix X
Melursus ursinus
Parictis montanus X
Tremarctos ornatus
Ursavus brevihinus X
Ursavus primaevus X
Ursus abstrusus X
Ursus americanus
Ursus arctos
Ursus maritimus
Ursus spelaeus X
Ursus thibetanus
Zaragocyon daamsi X
Ste
m
Be
ars
Cro
wn
Be
ars
Pandas
Tremarctinae
Brown Bears
Ursinae
Origin
Root(Total Group)
Crown
01020304050Y
pres
ian
Lute
tian
Bar
toni
an
Pria
boni
an
Rup
elia
n
Cha
ttian
Aqu
itani
an
Bur
diga
lian
Lang
hian
Ser
rava
llian
Tort
onia
n
Mes
sini
an
Zan
clea
nP
iace
nzia
nG
elas
ian
Cal
abria
nM
iddl
eU
pper
Eoc
ene
Olig
ocen
e
Mio
cene
Plio
cene
Ple
isto
cene
Hol
ocen
e
Paleogene Neogene Quat.
Ailuropoda melanoleuca
Tremarctos ornatus
Melursus ursinus
Ursus arctosUrsus maritimus
Helarctos malayanusUrsus americanusUrsus thibetanus
Parictis montanus XZaragocyon daamsi X
Ballusia elmensis XUrsavus primaevus X
Ursavus brevihinus X
Indarctos vireti XIndarctos arctoides X
Indarctos punjabiensis XAilurarctos lufengensis X
Agriarctos spp X
Kretzoiarctos beatrix X
Ursus abstrusus X
Ursus spelaeus X
Arctodus simus X
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Estimating EpidemiologicalParameters of an EbolaOutbreak
0175 255075100125150
Days
Course materials: http://phyloworks.org/resources/ssbws.html