Introduction to Bayesian Divergence Time Estimation

I  B
D T
E
Tracy Heath
Ecology, Evolution, & Organismal Biology
Iowa State University
@trayc7
http://phyloworks.org
SSB Workshop at Evolution 2015
Guarujá, Brazil

T-H M
What 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 for
the rate along the branch & the time duration of the
branch
• Sequence data alone are not informative for absolute
time (in years)
• To infer absolute times, additional data (e.g., fossils or
biogeography) are needed
• It’s very important to have a good understanding of all
data (including fossils) used for divergence-time
estimation
Course materials: http://phyloworks.org/resources/evol2015ws.html

B I
Estimate the probability of a hypothesis (model) conditional
on observed data.
The probability represents the researcher’s degree of belief.
Bayes’ Theorem speciﬁes the conditional probability of the
hypothesis given the data.

B’ T
Bayesian Fundamentals

B’ T
The posterior probability of a discrete parameter δ
conditional 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 all
possible values of δ.

B’ T
The posterior probability density a continuous parameter θ
conditional 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 all
possible values of θ.

E  P P
Once we have a model deﬁned that represents f(θ | D), how
do 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 computing
machines. 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 SSB
President—and his lecture on Bayesian phylogenetics
Slides source: 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
State
0 2500 5000 7500 10000 12500 15000 17500
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0
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(posterior)

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
State
0 2500 5000 7500 10000 12500 15000 17500
-6
-5
-4
-3
-2
-1
0
If step size is too
small, large-scale
trends will be
apparent
log(posterior)

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.TXT
State
0 2500 5000 7500 10000 12500 15000 17500
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-2
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(posterior)

M C M C (MCMC)
Thanks, Paul!
Slides source: https://molevol.mbl.edu/index.php/Paul_Lewis
See MCMCRobot, a helpful
software program for learning
MCMC by Paul Lewis
http://www.mcmcrobot.org

D T E
Goal: Estimate the branch lengths in units proportional to
time to understand the timing and rates of evolutionary
processes
Model how rates are
distributed across the tree
Describe the distribution of
speciation events over time
External calibration
information for estimates of
absolute 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)

A T-S  E
Phylogenetic trees can provide both topological information
and temporal information
100 0.020.040.060.080.0
Equus
Rhinoceros
Bos
Hippopotamus
Balaenoptera
Physeter
Ursus
Canis
Felis
Homo
Pan
Gorilla
Pongo
Macaca
Callithrix
Loris
Galago
Daubentonia
Varecia
Eulemur
Lemur
Hapalemur
Propithecus
Lepilemur
Mirza
M. murinus
M. griseorufus
M. myoxinus
M. berthae
M. rufus1
M. tavaratra
M. rufus2
M. sambiranensis
M. ravelobensis
Cheirogaleus
Simiiformes
Microcebus
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 of
evolutionary change is
constant over time
(branch lengths equal
percent sequence
divergence) 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 we
know the rate of change is
1% divergence per 10 My
A B C
20%
10%
10%
10%
200 My
400 My
200 My

T G M C
If we found a fossil of the
MRCA of B and C, we can
use it to calculate the rate
of change & date the root
of the tree
A B C
20%
10%
10%
10%
200 My
400 My

R  G M C
Rates 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 of
selection
• population sizes
10%
400 My
200 My
A B C
20%
10%
10%

U A
Sequence data provide
information about branch
lengths
In units of the expected # of
substitutions per site
branch length rate × time
0.2 expected
substitutions/site
PhylogeneticRelationshipsSequence
Data

R  T
The sequence data
provide information
about branch length
for any possible rate,
there’s a time that ﬁts
the branch length
perfectly
0
1
2
3
4
5
0 1 2 3 4 5
BranchRate
Branch Time
time = 0.8
rate = 0.625
branch length = 0.5
(based on Thorne & Kishino, 2005)

R  T
The expected # of substitutions/site occurring along a
branch is the product of the substitution rate and time
length = rate × time length = rate length = time
Methods for dating species divergences estimate the
substitution rate and time separately

B D T E
length = rate length = time
R (r,r,r,...,rN−)
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 branches
A Vector of internal node ages
θR,θA,θs Model parameters
D Sequence data
Ψ 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 V
Some models describing lineage-specific substitution rate
variation:
• Global molecular clock (Zuckerkandl & Pauling, 1962)
• Local molecular clocks (Hasegawa, Kishino & Yano 1989;
Kishino & Hasegawa 1990; Yoder & Yang 2000; Yang & Yoder
2003, Drummond and Suchard 2010)
• Punctuated rate change model (Huelsenbeck, Larget and
Swofford 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, Huelsenbeck
2012)
Models of Lineage-specific Rate Variation

R-C M
To accommodate variation in substitution rates
‘relaxed-clock’ models estimate lineage-speciﬁc substitution
rates
• Local molecular clocks
• Punctuated rate change model
• Log-normally distributed autocorrelated rates
• Uncorrelated/independent rates models
• Mixture models on branch rates

M R V
Are 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)
t2
t3
t4
t6
t7
t5
t8
t9
t10
tx
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
t1
Eocene Oligocene Miocene Plio Plei Hol
34 5.3 1.823.8 0.01
Epochs
Ma
Global expansion of C4 biomass
Major temperature drop and increasing seasonality
Faunal turnover
Krause et al., 2008. Mitochondrial genomes reveal an
explosive radiation of extinct and extant bears near the
Miocene-Pliocene boundary. BMC Evol. Biol. 8.
Taxa
1
5
10
50
100
500
1000
5000
10000
20000
0100200300
MYA
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 25.3
2. 0.081 * 25.5
3. 0.067 0.37 45.1
4. 0 * 3.1
Bg. 0.011 0.0011
OstariophysiAcanthomorpha
Teleostei
Santini et al., 2009. Did genome duplication drive the origin
of teleosts? A comparative study of diversification in
ray-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 important
for 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 distribution
on node times
f(A | θA)
Diﬀerent node-age priors make diﬀerent assumptions about
the timing of divergence events
Node Age Priors

S B P
Node-age priors based on stochastic models of lineage
diversiﬁcation
Constant-rate birth-death
process: at any point in
time a lineage can speciate
at rate λ or go extinct with
a rate of μ
Node Age Priors

S B P
Diﬀerent values of λ and μ lead
to diﬀerent trees
Bayesian inference under these
models can be very sensitive to
the values of these parameters
Using hyperpriors on λ and μ
accounts for uncertainty in these
hyperparameters
Node Age Priors

P  N T
Sequence data are only informative on relative rates & times
Node-time priors cannot give precise estimates of absolute
node ages
We need external information (like fossils) to calibrate or
scale the tree to absolute time
Node Age Priors

C D T
Fossils (or other data) are necessary to estimate absolute
node ages
There is no information in
the sequence data for
absolute time
Uncertainty in the
placement of fossils
A B C
20%
10%
10%
10%
200 My
400 My

C D
Bayesian inference is well suited to accommodating
uncertainty in the age of the calibration node
Divergence times are
calibrated by placing
parametric densities on
internal nodes oﬀset by age
estimates from the fossil
record
A B C
200 My
Density
Age

A F  C
Misplaced fossils can aﬀect node age estimates throughout
the 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 fossils
can provide minimum time
constraints for internal
nodes
Reliable maximum bounds
are typically unavailable
Minimum age
Time (My)
Calibrating Divergence Times

P D  C N
Common practice in Bayesian divergence-time estimation:
Parametric distributions are
typically oﬀ-set by the age
of the oldest fossil assigned
to a clade
These prior densities do not
(necessarily) require
speciﬁcation of maximum
bounds
Uniform (min, max)
Exponential (λ)
Gamma (α, β)
Log Normal (µ, σ2
)
Time (My)Minimum age

P D  C N
Calibration densities describe
the waiting time between
the divergence event and
the 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 node
ages are driven primarily by
the calibration density
Specifying appropriate
densities is a challenge for
most 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 to
eliminate the need for
ad hoc calibration
prior densities
Calibration densities
do not account for
diversiﬁcation 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 all
of the available fossils
Example: Bears
12 fossils are reduced
to 4 calibration ages
with calibration density
methods
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 are
part of the
diversiﬁcation process,
we can combine fossil
calibration with
birth-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 a
branching model that
accounts for
speciation, extinction,
and rates of
fossilization,
preservation, and
recovery
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 arbitrary
calibration densities
Better capture our statistical
uncertainty in species divergence dates
All reliable fossils associated with a
clade 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 specimens
provide historical
observations of the
diversiﬁcation process that
generated the tree of
extant 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 tree
and fossil observations
under a birth-death model
with rate parameters:
λ = speciation
μ = extinction
ψ = fossilization/recovery
150 100 50 0
Time

T F B-D P (FBD)
We use MCMC to sample
realizations of the
diversiﬁcation process,
integrating over the
topology—including
placement of the
fossils—and speciation times
0250 50100150200
Time (My)

I FBD T
Extensions of the fossilized birth-death process accommodate
variation in fossil sampling, non-random species sampling, &
shifts in diversiﬁcation rates.
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
Lower
Middle
Upper
Lower
Upper
Paleocene
Eocene
Oligocene
Miocene
Pliocene
Pleistocen
Jurassic Cretaceous Paleogene Neogene Q.
With character data for both fossil & extant species, we
account for uncertainty in fossil placement

D  C-P R
Analysis of morphology + DNA for fossil & extant taxa
Earlier age for crown MRCA is more consistent with the
fossil record
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
See Tanja Stadler's talk
on Tuesday at 13:30:
“A unified framework
for inferring phylogenies
with fossils''
(Figure adapted from Gavryushkina et al., arXiv:1506.04797)

S B-D P
A piecewise shifting model
where parameters change
over time
Used to estimate
epidemiological parameters
of an outbreak
0175 255075100125150
Days
(see Stadler et al. PNAS 2013 and Stadler et al. PLoS Currents Outbreaks 2014)

RB D: A S A
RevBayes – Fully integrative Bayesian inference of
phylogenetic parameters using probabilistic graphical models
and an interpreted language
http://RevBayes.com
Graphical model: Strict clock, pure birth process, GTR
sf
Q
[ fnGTR( ) ]
er_hp
1 1 1 1 1 1
er
phySeq
sf_hp
1 1 1 1
timetree
rho
0.068
root_time
38 50
extinction
0
speciation
10
clock_rate
2 4
phySeq.pInv
0
Example

G M  RB
Graphical models provide tools for
visually & computationally representing
complex, parameter-rich probabilistic
models
We can depict the conditional
dependence structure of various
parameters and other random variables
Höhna, Heath, Boussau, Landis, Ronquist, Huelsenbeck. 2014.
Probabilistic Graphical Model Representation in Phylogenetics.
Systematic Biology. (doi: 10.1093/sysbio/syu039)

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