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Lecture 6
Bayesian Inference and Molecular Dating
Thomas Bayes
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Bayesian Inference: The explanation with the highest posterior probability
Pr(H D) = Pr(H) Pr(D H)
Pr(D)
2. Bayes’ Theorem
Posterior probability, the probability of the hypothesis given the data
Prior probability, the probability of the hypothesis on previous knowledge
Likelihood function, probability of the data given the hypothesis
Unconditional probability of the data, a normalizing constant ensuring the posterior probabilities sum to 1.00
Pr(A and B) = Pr(A) Pr(B A) = Pr(B) Pr(A B) 1. Definition of conditional probability
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Pr(H D) = Pr(H) Pr(D H)
Pr(D)
Odds ratios are the simplest usage for Bayes’ theorem Felsenstein’s example: He gives his prior belief in “martians” Pr(H) = 1/4. A space probe sent to mars has probability of 1/3 of finding martians if they are there. It finds none, so Pr(D H) = 1/3, where H is that martians exist is 1/3
= 1/4 1/3
1 = 1/12
Assuming all the data is correct
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When the prior is a probability distribution, the posterior probability can be given as a probability distribution (often considered a density distribution under a curve)
0 24
Flat prior
0 24
Prior Skewed Prior
Parameter value Parameter value
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Influence of the Prior diminishes if the likelihood ratio between hypotheses increases with data collection
Post
erio
r
pheads
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
10 coin tosses, 3 heads 100 coin tosses, 30 heads
“heads” “tails”
pheads
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First use in phylogenetics: Li (1996, PhD thesis), Rannala and Yang (1996)A useful property of Bayesian inference for phylogenetics is that with flat priors (all hypotheses equal before the data is examined), posterior probabilities for two trees are proportional to their likelihood ratio.
Bayesian inference in phylogenetics
Tree1 Tree2 Tree3 Tree1 Tree2 Tree3 Tree1 Tree2 Tree3
Prio
r
Post
erio
r
Like
lihoo
d
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(ti D) = (ti) (D ti)
(tj) (D tj)
Where (D ti) = b (D ti,b, ) (b, ) dbd
TJ=1
Bayesian inference in phylogenetic notation
Posterior distribution
Prior distribution
Likelihood distribution
Summing over this integral becomes too complex with so many tree/parameter hypotheses
ti = Treei = substitution model b = branch-length
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Markov chain Monte Carlo (MCMC) analysis
Sampling the MCMC provides a valid approximation for the posterior distribution of trees (over 100,000s – 1,000,000s of generations) - without having to know the denominator
Tree 1Tree 2Tree 3
Generation: 1 2 3 4 5 6
New state acceptedNew state rejected
Bayesian Posterior Probability for Tree 1 (BPPtree 1)= 4/6
The chain
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Metropolis-Hastings algorithm (the proposal mechanism)
1. Start at a random (or predetermined) tree, Ti 2. Randomly select a tree that is a neighbour in tree space, the proposal tree, Tj 3
2e.g.
541
4
2 531
3. Compute the acceptance ratio R = Pr(Tj) Pr(D Tj )
Pr(Ti) Pr(D Ti )4. If R ≥ 1, accept Tj
5. If R < 1, randomly draw from 0-1, if < R, accept Tj
6. Otherwise reject Tj and keep Ti 7. Return to step 2
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• Given an efficient proposal mechanism and sufficient generations, the MCMC with reach an equilibrium distribution
• The problem of summing across all hypotheses cancels out in the acceptance ratio for Tj / Ti
• The section of the MCMC that is “finding its way” towards the equilibrium distribution (the burnin) can be discarded - it is not a valid approximation for the posterior probability
Useful properties of MCMC for Bayesian inference
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Metropolis-coupled MCMC (MC3)
n chains, of which n-1 are “heated” such that they can more easily move across peaks and valleys in the landscape of trees.
After all n chains have gone one step a swap between randomly chosen chains is proposed (in much the same way as between generations). If accepted the two chains switch states.
The cold chain (which could be stuck in a local optimum) can escape when a proposed swap with a hot chain is successful.
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4100 -- (-15395.901) (-15376.724) [-15367.235] 4200 -- (-15395.854) (-15377.874) [-15368.146] 4300 -- (-15381.440) [-15376.072] (-15368.164) 4400 -- (-15375.695) (-15369.686) [-15369.127] 4500 -- [-15374.458] (-15372.051) (-15370.876) 4600 -- [-15358.899] (-15366.374) (-15365.927) 4700 -- [-15350.419] (-15366.082) (-15366.168) 4800 -- (-15369.614) [-15368.152] (-15369.902)
Sampling three simultaneous chains (MrBayes 3.0: Heulsenbeck and Ronquist, 2001)
Generation chain 1 chain 2 chain 3
Proposing new trees along individual chains and swapping trees between chains (to more efficiently explore tree space)
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Inference is based only on sampling from the cold chain - for which the acceptance ratio for changes between generations is appropriate for approximating the posterior probability.
tree tree_1 [p = 0.455, P = 0.455](Ostrich,Rhea,(Moa,(Kiwi,(Emu,Cassowary))))tree tree_2 [p = 0.215, P = 0.670] (Ostrich,(Kiwi,(Moa,Rhea)),(Emu,Cassowary))tree tree_3 [p = 0.118, P = 0.788] (Ostrich,Moa,(Rhea,(Kiwi,(Emu,Cassowary)))tree tree_4 [p = 0.081, P = 0.869] (Ostrich,(Moa,Rhea),(Kiwi,(Emu,Cassowary)))tree tree_5 [p = 0.050, P = 0.919] (Ostrich,Kiwi,(Moa,(Rhea,(Emu,Cassowary))))
From Phillips, unpub.
Ostrich
Emu
Moa
Rhea
Kiwi
Cassowary
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H1: AnapsidaH3: ArchosauriaH2: Diapsida
Amphibia (outgroup)
Squamata
Aves
CrocodiliaMammalia
A Return to the Turtle example
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Bayesian inference in MrBayes 3.0
• The same data matrix as for the likelihood example (16 taxa, 3110 nucleotides)
• GTR+I+ model (AIC recommendation in ModelTest)
• Four MC3 chains (3 are “hot”): 2,000,000 generations
• sample every 1000th tree from the “cold” chain
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+------------------------------------------------------------+ -25419.04 | *****************************************************| | ** | | | | * | | * | | * | | | | | | * | | | | | | | |* | | | | | +------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ -33441.42 ^ ^ 1 10000
Estimated marginal likelihood = -25423.76 (arithmetic mean) = -25451.08 (harmonic mean)
Discard the 1st 250,000 generations as the “burnin”
2,000,000generations
likel
ihoo
d
Equilibrium distribution
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95% Cred. Interval -------------------- Parameter Mean Variance Lower Upper Median -------------------------------------------------------------- TL 2.513384 0.005489 2.373000 2.663000 2.512000 r(G<->T) 1.000000 0.000000 1.000000 1.000000 1.000000 r(C<->T) 14.895438 2.952511 11.968691 18.597121 14.745678 r(C<->G) 0.088475 0.004142 0.011433 0.241428 0.074255 r(A<->T) 2.489403 0.105833 1.952249 3.220099 2.453488 r(A<->G) 8.309381 0.935429 6.684136 10.510313 8.229202 r(A<->C) 2.481878 0.108561 1.928505 3.190486 2.453910 pi(A) 0.323330 0.000044 0.310754 0.336207 0.323446 pi(C) 0.215915 0.000035 0.204413 0.227673 0.216025 pi(G) 0.209200 0.000035 0.197702 0.221016 0.209162 pi(T) 0.251555 0.000037 0.239747 0.263308 0.251377 alpha 1.017169 0.015644 0.785668 1.261082 1.013706 pinvar 0.226459 0.000787 0.164235 0.274568 0.229746 --------------------------------------------------------------
Model parameter summaries for the 1750 samples from the MCMC chain
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Unlike likelihood, for which parameters are optimised, Bayesian inference gives a Posterior density distribution for parameter values (integrated over all trees sampled from the chain)
0 0.2 0.4
freq
uenc
y
Proportion of invariant sites
95% credible interval
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LnL -25423.738 1232.859TL 2.513 1242.35r(G<->T) 1 -r(C<->T) 14.895 586.829r(C<->G) 8.846E-2 976.73r(A<->T) 2.49 613.916r(A<->G) 8.31 605.106r(A<->C) 2.482 608.116pi(A) 0.323 1269.731pi(C) 0.216 1235.234pi(G) 0.209 1311.202pi(T) 0.252 1069.327alpha 1.017 1592.408pinvar 0.226 1607.974
Statistic Mean Estimated sample size
Sampling analysis (e.g. Tracer 1.0) Rambaut and Drummond
All ESS should be at least >100
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Has the MCMC run for sufficient generations?
1. Have the chains converged on similar likelihood values and swaps being made between each of them?
2. Do different runs of the same analyses converge?
3. Sampling over generation plots reveal that likelihoods have levelled (equilibrium distribution is reached)
4. Estimated sample sizes for all parameters are >100, indicating substantial coverage of parameter space
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SalamanderCaecilian
SkinkIguana
1.00
Green Turtle
Painted Turtle1.00
Alligator
Caiman1.00
CassowaryPenguin
1.000.99
1.00
1.00
PlatypusEchidna
1.00
Dog3 toed Sloth
1.00
KangarooOpossum
1.000.79
1.00
1.00
Arc
hosa
uria
Dia
psid
iaM
amm
alia
Amphibia
Bayesian inference tree with Posterior probabilities
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Birds
Crocodilia
Amphibia
Turtles
Squamates
Mammalia
Birds
Crocodilia
Amphibia
Turtles
Squamates
Mammalia
Birds
Crocodilia
Amphibia
Mammalia
Squamates
Turtles
Tree 1
Tree 2
Tree 3
Tree1 Tree2 Tree3-lnL +36.1 +11.7 <best>KH 0.002 0.044 --SH 0.003 0.153 --AU 0.001 0.054 --NPBP 0.005 0.041 --SOWH <0.001 <0.001 --BPP <0.001 <0.001 --
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Controversies with Bayesian inference
1. Flat prior? It depends on perspective
0.0 0.25 0.5 0.75
Branch-length as the probability of change
0 1 2 3 4 5
Priors for the same branch lengths (under a Jukes-Cantor model)
Branch-length as expected substitutions per site
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2. A flat prior on branch-lengths is not a flat prior on clade probabilities
A flat prior across all topologies for the whole tree favours the largest and smallest clades over clades of intermediate taxon inclusion
However, the influence on posterior probabilities maybe quite small
Pickett and Randle (MPE, 2005)
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3. Bounds on priors
0.0 100
0.0 100
Branch-length
Branch-length
Prio
rPo
ster
ior
95% credible interval
A zero to infinity “bound” provides a worst case scenario
MLE
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Simmons et al. (MBE, 2004)
4. Bayesian posterior probabilities tend to overestimate clade support (underestimate sampling effects) when the substitution model is misspecified – Which is essentially always when using biological datasets
True
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Bayesian inference methods are now being incorporated into phylogenetic analysis of morphological data
Right scapula of an echidna (a.) and a marsupial cat (b) in distal view. ap, acromion process; gl, glenoid; if, infraspinous fossa; mc, metacoracoid; sf, supraspinous fossa; sp, scapular spine; vpg, ventral process of glenoid.
a. 0 0 1 0 0
b. 0 1 1 1 1
mc
fuse
d to
scap
ular
glen
oid
(gl)
orie
ntat
ion
Supr
aspi
nous
fos
sa (s
f)ap
rela
tive
to g
l
vpg
poin
ts v
ente
ral
gl
glif
if
ap ap
mc
mcsfsp
sp
vpg
vpg
medialmedial
(b.)(a.)
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The Assumption for Likelihood that sites evolve by common mechanisms is not clear for morphological dataThe state “1” for one character does not mean the same thing as state “1” for another character
Oak ATGACCGCTGCCAG Ash ACGCTCGCCATCAG Maple ATGCTCGCTACCGG
Transitions at six sites, only one transversion is observed
An ML model would allow for different transition and transversion substitution rates
a. 0 0 1 0 0
b. 0 1 1 1 1
mc
fuse
d to
scap
ular
glen
oid
(gl)
orie
ntat
ion
Supr
aspi
nous
fos
sa (s
f)ap
rela
tive
to g
l
vpg
poin
ts v
ente
ral
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Lewis (Syst. Biol., 2001) recognized that ML estimation of branch-lengths could help interpret the nature of morphological change among taxa (e.g. as resulting from shared ancestry or convergence)
Mkv model for morphology resembles Jukes-Cantor (JC):
• All states assumed to have the = equilibrium frequency
• Changes between all states are equally probable
Differences:
• Multiple (>4) states allowed
• conditional on no characters being constant across all taxa
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The root of the Australidelphian marsupial tree
dasyurids
bandicoots
diprotodontsS. American marsupials
Aus
trala
sian
M
arsu
piar
nivo
ra
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dasyurids
bandicoots
diprotodonts
S. American marsupials
syndactylus
Non-syndactylus
Syndactyla
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Phillips et al. (Syst. Biol, 2006)
ML, MP and Bayesian inference on 17,864 nuclear and mitochondrial nucleotide sites
Aus
trala
sian
M
arsu
pica
rniv
ora
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Horovitz and Sánchez-Villagra (Cladistics, 2002)
Parsimony on 230 morphological characters
Syndactyla + Dromiciops
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MayulestesPucadelphys
Shrew opossumMonito del monte
Pygmy possumSugar gliderRingtail possum
CuscusBrushtail possum
Tree kangarooPademellonKangaroo
Forest wallabyWombatKoala
MulgaraQuollPhascogale
Marsupial moleSpiny bandicootLong-nosed bandicoot
Virginia opossumGrey Short-tailed opossum
Bayesian inference (Mkv-model) on the 230 morphological characters of Horovitz and Sánchez-Villagra (2002)
Australasian Marsupicarnivora
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Combined Molecular and Morphological data analysis
Allows model-based estimation for molecular sequences to be combined with morphological data (including for fossil taxa)
Taxon1 ACGTAAGTC 0000110 Taxon2 ATGGAAATT 1110302 Taxon3 ACATAAATC 1020111 Taxon4 ACGCTAGTC 0010012
Partition 1 (e.g. GTR model)
Partition 2 (Mkv model)
(ti D12) = (ti) (D1 ti) (D2 ti)
(tj) (D1 tj) (D2 tj)TJ=1
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Glenner et al. (Curr. Biol., 2005)
Deuterostomes (inc. echinoderms, sea squirts, vertebrates)
Ecdysozoa (inc. arthropods, nematodes)
(Platyhelminthes, rotifers, acorn worms)
Lophotrochozoa (inc. molluscs, annalids)
Combined (DNA/morphology) Bayesian analysis of Metazoans
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Bayesian character state reconstruction: Shell hairs on the Trochulus snails
Pfenninger et al. (BMC Evol. Biol., 2005)
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Relaxed-clock Bayesian Phylogeneticsr = rate (substitutions/site/time); t = node height; branch-length (b) = rt; n = number of taxa
A. Clock B. No-clock C. Relaxed-clock
t1ri
b4
b3b2b1 b1 b5
b4
b7b3
b2
b6
t3rk
t4rl
t2rj
Forces a molecular clock (r is constant) – assumes deviation from this is sampling error.
Different rate for each branch – no sampling error is assumed.
Rates are influenced by sampling error and vary according to an underlying distribution (e.g. exponential or lognormal).
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A. Clock B. No-clock C. Relaxed-clock
t1ri
b4
b3b2b1 b1 b5
b4
b7b3
b2
b6
t3rk
t4rl
t2rj
n-1 branch-length parameters
2n-3 branch-length parameters
n-1 node heights and 1-2 rate distribution parameters
Most sequence datasets have not evolved in a clock-like fashion, and so the assumption of a clock often produces incorrect tree topologies
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• By allowing for the influence of sampling error, relaxed-clock models can more accurately infer underlying substitution rates and hence provide greater phylogenetic accuracy.
• By estimating fewer branch-length parameters, less variance will tend to be associated with relaxed-clock model estimates, thus providing for greater phylogenetic precision than no-clock models
• However, we need to know how well fitting the rate distribution models are
Relaxed-clocks : the holy grail of phylogenetics?
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A.
B. Yeasts (106)C. Plants (61)D. Animals (99)E. Primates (500)
B.C.D.E.
A. Bacteria (102)
Drummond et al. (PLoS Biol., 2006)
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Drummond et al. (PLoS Biol., 2006)
Relaxed phylogenetics allows co-estimation of phylogeny and divergence timing