Bayesian Phylogenetics:an introduction

Marc A. Suchardmsuchard@ucla.edu

UCLA

Who is this man?

How sure are you?

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The one ‘true’ tree?

Methods we’ve learned so fartry to find a single tree thatbest describes the dataHowever, they do not searcheverywhere, and it is difficultto find the “best” treeMany (gazillions of) trees maybe almost as good

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Bayesian phylogenetics: general principle

Using Bayesian principles, we will search for and average over setsof plausible trees (weighted by their probability) instead a single“best” treeIn this method, the “space” that you search is limited by priorinformation and the data

The posterior distribution oftrees naturally translates intoprobability statements (anduncertainty) on aspects ofdirect scientific interest

I When did an evolutionaryevent happen?

I Are a subset of sequencesmore closely related?

The cost: we must formalizeour prior beliefs

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Conditional probability: intuition

Philippe is a hipster.Philippe is a hipster

and rides a single-speed bike.

Which is more probable?

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Conditional probability: intuition

Arbitrary events A (hipster) and B (bike) from sample space U

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Bayes theorem

Definition of conditional probability in words:

probability(A and B) = probability(A given B)× probability(B)

In usual mathematical symbols:

p(A|B)p(B) = p(A,B) = p(B|A)p(A)

With a slight re-arrangement:

p(A|B) =p(B|A)p(A)

p(B)

“Just" a restatement of conditional probability

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Bayes theorem

Integration (averaging) yields a marginal probability:

p(A) =

∫p(A,B)dB =

∫p(A|B)p(B)dB︸ ︷︷ ︸

over all possible values of B

probability(hipster) = probability(hipster and has bike) +probability(hipster and has no bike)

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Conditional probability: pop quiz

What do you know about Thomas Bayes?Bayes theorem?

Some discussion points:Favorite game? Best buddies?

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Bayes theorem for statistical inference

Unknown quantity θ (model parameters, scientific hypotheses)Prior p(θ) beliefs before observed data Y become availableConditional probability p(Y |θ) of the data given fixed θ – alsocalled the likelihood of YPosterior p(θ|Y ) beliefs:

p(θ|Y ) =p(Y |θ)p(θ)p(Y )

p(θ) and p(Y |θ) – easyp(Y ) =

∫p(Y |θ)p(θ)dθ – hard

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Bayesian phylogenetic inference

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Bayesian phylogenetic inference

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Bayesian phylogenetic inference

Posterior:

p(θ|Y ) =p(Y |θ)p(θ)p(Y )

Trouble: p(Y ) is notcomputable – sum over allpossible treesFor N taxa: there are G(N) =(2N − 3)× (2N − 5)× · · · × 1

θ = (tree, substitution process)p(Y |θ) - continuous-timeMarkov chain process thatgives rise to sequences at tipsof tree

E.g., G(21) > 3× 1023

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Priors

Strongest assumption: most parameters are separable, e.g. the treeis independent of the substitution processWeaker assumption: tree ∼ Coalescent processWeaker assumption: functional form on substitution parameters

Specialized priors as wellIf worried: check sensitivity

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Posterior inference

Numerical (Monte Carlo) integration as a solution:

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Markov chain Monte Carlo

Metropolis et al (1953) andHastings (1970) proposed astochastic integrationalgorithm that can explore vastparameter spacesAlgorithm generates a Markovchain that visits parametervalues (e.g., a specific tree)with frequency equal to theirposterior density / probability.Markov chain: random walkwhere the next step onlydepends on the currentparameter state

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Metropolis-Hastings Algorithm

Each step in the Markov chain starts at its current state θ andproposes a new state θ? from an arbitrary proposal distributionq(·|θ) (transition kernel)θ? becomes the new state of the chain with probability:

R = min

p(θ?|Y )

p(θ|Y )× q(θ|θ?)q(θ?|θ)

= min

p(Y |θ?)p(θ?) / p(Y )

p(Y |θ)p(θ) / p(Y )× q(θ|θ?)q(θ?|θ)

= min

p(Y |θ?)p(θ?)

p(Y |θ)p(θ)× q(θ|θ?)q(θ?|θ)

Otherwise, θ remains the state of the chain

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Posterior samplingWe repeat the process of proposing a new state,calculating the acceptance probability and eitheraccepting or rejecting the proposed move millions oftimes

Although correlated, the Markovchain samples are valid draws fromthe posterior; however . . .

Initial sampling (burn-in) is oftendiscarded due to correlation withchain’s starting point ( 6= posterior)

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Transition Kernels

Often we propose changes to only a small # of dimensions in θ at atime (Metropolis-within-Gibbs)In phylogenetics, mixing (correlation) in continuous dimensions ismuch better (smaller) than for the treeSo, dominate approach has been keep-it-simple-stupid –alternatives exist and may become necessary:

I Gibbs sampler; slice sampler; Hamiltonian MC

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Tree Transition Kernels

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Posterior Summaries

For continuous θ, consider:posterior mean or median ≈MCMC sample average ormedianquantitative measures ofuncertainty, e.g. high posteriordensity interval

Credible Regions

Parameter (x)

The Bayesian equivalent of a confidence interval is called the highest posterior density (HPD) credible region. This is the

smallest region that contains 95% of the posterior probability.

lower 95% HPD limit

post

erio

r pr

obab

ility

di

stri

butio

n

upper 95% HPD limit

For trees, consider:scientifically interestingposterior probabilitystatement, e.g. the probabilityof monophyly ≈ MCMCsample proportion under whichhypothesis is true

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Posterior Probabilities

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Summarizing Trees

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MCMC Diagnostics: within a single chain

VisuallyinspectMCMCoutput traces

Measure au-tocorrelationwithin achain: theeffectivesample size(ESS)

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MCMC Diagnostics: across multiple chains

VisuallyinspectMCMCoutput traces

Comparing different chains → variance among and between chains

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Improving Mixing

(Only if convergence diagnostics suggest a problem)

Run the chain longerUse a more parsimonious model(uninformative data)Change tuning parameters of transitionkernels to bring acceptance rates to 10% to70%

Use different transition kernels (consult anexpert)

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Improving Mixing

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Why Bother being Bayesian?

In practice, we have almost no prior knowledge for the modelparameters. So, why bother with Bayesian inference?

Analysis provides directly interpretable probability statementsgiven the observed dataMCMC is a stochastic algorithm that (in theory) avoids gettingtrapped in local sub-optimal solutionsSearch space under Coalescent prior is astronomically “smaller”By numerically integrating over all possible trees, we obtainmarginal probability statements on hypotheses of scientific interest,e.g. specific branching events or population dynamics, avoiding bias

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