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7. Bayesian phylogenetic analysis using MrBAYES

USTJeong Dageum2010.05.24

Thomas Bayes(1702-1761)

The Phylogenetic Handbook – Section III, Phylogenetic inference

Prior * LikelihoodPosterior

Normalizing constant

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7. 1 Introduction7.2 Bayesian phylogenetic inference7.3 Markov chain Monte Carlo sampling7.4 Burn-in, mixing and convergence7.5 Metropolis coupling7.6 Summarizing the results7.7 An introduction to phylogenetic models7.8 Bayesian model choice and model averaging7.9 Prior probability distribution

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Next year’s world championships in ice hockey? Sweden?!!!!

15 years1 of 7 countries1:7 or 0.14

2 gold medal2:15 or 0.13

Final? Semifinal?

Russia, Canada, Finland, Czech Republic, Sweden, Slovakia, United States: 7

7.1 Introduction

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Bayesian approach: Bayesian inference is just a mathematical formalization of a decision process that most of us use without reflecting on it

Posterior

Prior * Likelihood

Normalizing Constant

7.1 Introduction

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?

? Forward probability

50: 50 ?

W ball proportion : PB ball proportion: 1-p

7.1 Introduction

a, b

Converse !

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f(p |a,b) = ? : Reverse probability problem

We know a and b, thenWhat is the probability of a particular value of p?

Need Prior beliefs about the value of p

7.1 Introduction

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[Probability mass function]: a function describing the probability of a discrete Random variable (ex: Dice)

Box 7.1 Probability distributions – [Considering Prior]

[Probability density function]: For a continuous variable, the equivalent functionThe value of this function is not a probability

Exponential distribution: A better choice for a vague prior on branch lengths

7.1 Introduction

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Gamma distribution: 2 parameters (shape parameter α, scale parameter β

Small value of α: the distribution is L-shapedAnd the variance is largeHigh value of α: similar to normal distribution

7.1 IntroductionBox 7.1 Probability distributions – [Considering Prior]

The beta distribution denoted Beta (α1, α2)Describes the probability on two proportions, which are associated with the weight parameters.

The beta distribution

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Posterior probability distribution

7.1 Introduction

How do we calculate f(a,b)?

Bayes’ theoremWe can calculate f(a,b|p)We can specify f(p)

To integrate over All possible values of p

- > Denominator is a normalizing constant

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7.2 Bayesian phylogenetic inference

P(Tree |Data) = P (Data)

P(Data |Tree)P (Tree)Posterior

Likelihood * Prior

Normalizing constant

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X: the matrix of aligned sequences Θ: topology, branch length, model.. Θ = (τ: topology parameter υ: branch lengths on the tree) substitution model parameters to be considered

7.2 Bayesian phylogenetic inference

X(Data) : fixed, Θ(parameter): Random

(Jukes Cantor substitution model)

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7.2 Bayesian phylogenetic inference

Parameter spaceIt corresponds to a particular set of branch lengths on that topology

Summarized all joint probabilities along one axis of the table, we obtain the marginal probabilities for the corresponding parameter.

Each cell

[Bayesian inference: there is no need to decide on the parameters of interest before performing the analysis]

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7.3 Markov chain Monte Carlo sampling

[For parameter sampling]

Markov chain Monte Carlo steps1. Start an arbitrary point (θ)2. Make a small random move (to θ*)3. Calculate height ration (r) of new state (to θ*) to old state (θ)

(a) r>1: new state accepted(b) r<1: new state accepted with probability r

if new state rejected, stay in old state4. Go to step 2

f (θ | θ*)

(Prior ratio)

)

(likelihood ratio)

(proposal ratio)

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7.3 Markov chain Monte Carlo samplingBox 7.2 Proposal mechanism – To change continuous variables

1) Proposal step2) Acceptance/Rejection step

[Sliding window proposal]

[Normal proposal] (similar to the above one)

[The beta and Dirichlet proposal]

ω: tuning parameterLarge: more radical proposal & lower acceptance ratesSmall: more modest changes & higher acceptance rate

σ2: Determine how drastic the new proposals are andhow often they will be accepted

[Multiplier proposal]

σ2: Determine how drastic the new proposals are andhow often they will be accepted

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Discard

7.4 Burn-in, mixing and convergence – [about performance of an MCMC run]

* Trace plot* To confirm convergence* Mixing behavior

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7.4 Burn-in, mixing and convergenceThe mixing behavior of a Metropolis sampler can be adjusted using its tuning parameter

Poor mixing

Poor mixing

Good mixing

ω is too small,The proposal will be accepted wellTakes long time to cover the all region

ω is too large,The proposal will be rejected wellTakes long time to cover the all region

ω is an intermediate value,Moderate acceptance rates

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7.4 Burn-in, mixing and convergence

Convergence diagnostics help determine the quality of a sample from the posterior.3 different types of diagnostics

(1) Examining autocorrelation times, effective sample sizes, and other measures of the behavior of single chains

(2) Comparing samples from successive time segments of a single chain

(3) Comparing samples from different runs.

=> In Bayesian MCMC sampling of phylogenetic problems, the tree topology is typically the most difficult parameter to sample from

ÞThe approach to solve this problem is to focus on split frequencies instead.•A split is a partition of the tips of the tree into two non-overlapping sets;•To calculate the average standard deviation of the split frequencies.

ÞPotential Scale Reduction Factor(PSRF)•PSRF compares the variance among runs with the variance within runs.•As the chains converge, the variances will become more similar and the PSRF will approach 1.o

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7.5 Metropolis coupling – [To activate the mixing]Cold chain, Hot chain•When: Difficult of impossible to achieve convergence•Metropolis coupling: A General technique to improve mixing

* An incremental heating scheme T = 1/ 1 + λi

where i∈{ 0,1,…k} for k heated chains, with i=0 for the cold chain, and λ is the temperature factor ( intermediate value of λ works best)

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7.6 Summarizing the results

Stationary phase of the chain/ Adequate sample> To compute an estimate of the marginal posterior distribution> Summarized using statistics* Bayesian statisticians : 95% credibility interval.

The posterior distribution on topology and branch lengths is more difficult to summarize efficiently.

* To illustrate the topological variance in posterior-> Estimated number of topologies in various credible sets.

* To give the frequencies of the most common splits=> A majority rule consensus tree

*The sampled branch lengths are even more difficult to summarize adequately.

ÞTo display the distribution of branch length values separatelyÞTo pool the branch length samples that correspond to the same split

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7.7 An introduction to phylogenetic modelsPhylogenetic model:

1) A Tree model • Unrooted / rooted model, • Strict / relaxed clock tree model

2) A substitution modelThe substitution model, Q matrices

The general time-reversible(GTR) model

* Factor πi: corresponds to the stationary state frequency of the receiving state* Factor rij,: determines the intensity of the exchange between pairs of states, controlling for the stationary state frequencies

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7.8 Bayesian model choice and model averaging

Prior Bayes factor

The probability of the data given the chosen model after we have integrated out all parameters: normalizing constant ( model likelihood)

* Bayes’ theorem

* Bayes factor comparisons are truly flexible. - Unlike likelihood ratio tests, No requirement for the models to be nexted- Unlike Akaike Information Criterion, Bayesian Information Criterion(confusingly named) no need to correct for the number of parameters in the model.

To estimate the model likelihood-> Use harmonic means in the MCMC run.

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7.9 Prior probability distributions – cautionary notes.

The priors : negligible influence on the posterior distribution

The Bayesian approach typically handles weak data quite well.

But when the data are weak, Extremely low likelihoods that attract the chain. ;;;

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Schematic overview of the models implemented in MrBayes3. Each box gives the available settings in normal font and then the program commands and coommand options needed to invoke those settings in italics

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PRACTICE

7.10 Introduction to Mrbayes7.10.1 Acquiring and installing the program7.10.2 Getting started7.10.3 Changing the size of the Mrbayes window7.10.4 Getting help

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PRACTICE

7.11 A simple analysis7.11.1 Quick start version7.11.2 Getting data into Mrbayes7.11.3 Specifying a model7.11.4 Setting the priors7.11.5 Checking the model7.11.6 Setting up the analysis7.11.7 Running the analysis7.11.8 When to stop the analysis7.11.9 Summarizing samples of substitution model parameters7.11.10 Summarizing samples of trees and branch lengths

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PRACTICE

7.12 Analyzing a partitioned data set simple analysis7.12.1 Getting mixed data into Mrbayes7.12.2 Dividing the data into partitions7.12.3 Specifying a partitioned model7.12.4 Running the analysis7.12.5 Some practical advice