Lecture 6

Bayesian Inference and Molecular Dating

Thomas Bayes

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

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

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

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

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

(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

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

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

• 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

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.

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)

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

H1: AnapsidaH3: ArchosauriaH2: Diapsida

Amphibia (outgroup)

Squamata

Aves

CrocodiliaMammalia

A Return to the Turtle example

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

+------------------------------------------------------------+ -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

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

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

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

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

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

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 --

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

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)

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

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

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.)

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

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

The root of the Australidelphian marsupial tree

dasyurids

bandicoots

diprotodontsS. American marsupials

Aus

trala

sian

M

arsu

piar

nivo

ra

dasyurids

bandicoots

diprotodonts

S. American marsupials

syndactylus

Non-syndactylus

Syndactyla

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

Horovitz and Sánchez-Villagra (Cladistics, 2002)

Parsimony on 230 morphological characters

Syndactyla + Dromiciops

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

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

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

Bayesian character state reconstruction: Shell hairs on the Trochulus snails

Pfenninger et al. (BMC Evol. Biol., 2005)

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).

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

• 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?

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)

Drummond et al. (PLoS Biol., 2006)

Relaxed phylogenetics allows co-estimation of phylogeny and divergence timing