Introduction to Bayesian Divergence Time Estimation

I BD TETracy Heath

Ecology, Evolution, & Organismal BiologyIowa State University

@trayc7http://phyloworks.org

SSB Workshop at Evolution 2015Guarujá, Brazil

https://twitter.com/trayc7

http://phyloworks.org/

T-H MWhat I hope to emphasize here:• Bayes’ theorem is a beautiful thing• The substitution rate & time are confounded parameters• To estimate branch time we need separate models forthe rate along the branch & the time duration of thebranch

• Sequence data alone are not informative for absolutetime (in years)

• To infer absolute times, additional data (e.g., fossils orbiogeography) are needed

• It’s very important to have a good understanding of alldata (including fossils) used for divergence-timeestimation

Course materials: http://phyloworks.org/resources/evol2015ws.html

http://phyloworks.org/resources/evol2015ws.html

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.

http://bit.ly/1Jz0Ta7

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.


http://en.wikipedia.org/wiki/Marginal_distribution

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.


http://en.wikipedia.org/wiki/Marginal_distribution

E P P

Once we have a model defined that represents f(θ | D), howdo we compute the posterior probability?

f(θ | D) =f(D | θ)f(θ)∫

θ f(D | θ)f(θ)dθ


M C M C (MCMC)

An algorithm for approximating the posterior distribution

Metropolis, Rosenbluth, 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.


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


https://molevol.mbl.edu/index.php/Paul_Lewis

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


(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)


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


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

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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)



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

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If step size is too small, large-scale trends will be apparentlo

g(po

ster

ior)



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

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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


https://molevol.mbl.edu/index.php/Paul_Lewis

http://www.mcmcrobot.org

D T E

Goal: Estimate the branch lengths in units proportional totime to understand the timing and rates of evolutionaryprocesses

Model how rates aredistributed across the tree

Describe the distribution ofspeciation events over time

External calibrationinformation for estimates ofabsolute node times

Paleocene Eocene

102030405060 0

Oligocene Miocene Po Ps

Paleogene Neogene Qu.

Age (Ma)

MRCA of extant

penguins

Eudyptes

Megadyptes

Aptenodytes

Pygoscelis

Spheniscus

Eudyptula

Icadyptes salasi

Waimanu manneringi

Spheniscus muizoni

Palaeospheniscus

patagonicus

Kairuku waitaki

(Figure adapted from Gavryushkina et al., arXiv:1506.04797)

http://arxiv.org/abs/1506.04797

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


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


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

Phyl

ogen

etic

Rel

atio

nshi

psSe

quen

ceD

ata

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

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

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

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

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

• Global molecular clock• Local molecular clocks• Punctuated rate change model• Log-normally distributed autocorrelated rates• Uncorrelated/independent rates models• Mixture models on branch rates

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

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

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

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)

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


P D C N

Calibration densities describethe waiting time betweenthe divergence event andthe age of the oldest fossil

Minimum age

Exponential (λ)

Time (My)


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


Ursavus primaevus

Agriarctos spp.


Indarctos vireti

Indarctos arctoides



Cave bear


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


Ursavus primaevus

Agriarctos spp.


Indarctos vireti

Indarctos arctoides



Cave bear


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


Ursavus primaevus

Agriarctos spp.


Indarctos vireti

Indarctos arctoides



Cave bear


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


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)


I FBD TExtensions of the fossilized birth-death process accommodatevariation in fossil sampling, non-random species sampling, &shifts in diversification rates.

0102030405060708090100

110

120

130

140

150

160

170

180

190

200

Lowe

r

Midd

le

Upper

Lowe

r

Upper

Paleo

cene

Eocene

Oligo

cene

Mioc

ene

Plioc

ene

Pleis

tocen

Jurassic Cretaceous Paleogene Neogene Q.

With character data for both fossil & extant species, weaccount for uncertainty in fossil placement

D C-P RAnalysis of morphology + DNA for fossil & extant taxaEarlier age for crown MRCA is more consistent with thefossil record

Paleocene Eocene

102030405060 0

Oligocene Miocene Po Ps

Paleogene Neogene Qu.

Age (Ma)

MRCA of extantpenguins

Eudyptes

Megadyptes

Aptenodytes

Pygoscelis

Spheniscus

Eudyptula

Icadyptes salasi

Waimanu manneringi

Spheniscus muizoni

Palaeospheniscuspatagonicus

Kairuku waitaki

See Tanja Stadler's talkon Tuesday at 13:30: “A uni�ed framework for inferring phylogenies with fossils''

(Figure adapted from Gavryushkina et al., arXiv:1506.04797)

http://arxiv.org/abs/1506.04797

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)

RB D: A S ARevBayes – Fully integrative Bayesian inference ofphylogenetic parameters using probabilistic graphical modelsand an interpreted languagehttp://RevBayes.comGraphical model: Strict clock, pure birth process, GTR

sf

Q[ fnGTR( ) ]

er_hp1 1 1 1 1 1

er

phySeq

sf_hp1 1 1 1

timetree

rho0.068

root_time

38 50

extinction0

speciation

10

clock_rate

2 4

phySeq.pInv0

Example

http://revbayes.github.io/

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)

http://sysbio.oxfordjournals.org/content/early/2014/07/26/sysbio.syu039

Introduction to Bayesian Divergence Time Estimation

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