lecture12.pdf Introduction to bioinformatics

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Introduction to Bioinformatics for
Computer Scientists
Lecture 12

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Exam
● Exam days:
● Feb 10 only for those who can't make the dates in April!
● April 22, 23, and 24
● I just sent around a doodle
● Also register for exam via campus.kit.edu !

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Plan for next lectures
● Today
● Bayesian statistics & (MC)MCMC methods
● Advanced MCMC
● Population genetics
● Course & Exam review

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Outline for today
● Bayesian statistics
● Monte-Carlo simulations
● Markov-Chain Monte-Carlo (MCMC) methods
● Metropolis-coupled MCMC-methods
● Course beers tonight ! 19:00 at Vogelbräu Karlsruhe

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Bayesian and Maximum Likelihood
Inference
● In phylogenetics Bayesian and ML (Maximum Likelihood)
methods have a lot in common
● Computationally, both approaches re-evaluate the phylogenetic
likelihood over and over and over again for different tree
topologies, branch lengths, and model parameters
● Bayesian and ML codes spend approx. 80-95% of their total run
time in likelihood calculations on trees
● Bayesian methods sample the posterior probability distribution
● ML methods strive to find a point estimate that maximizes the
likelihood

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Bayesian Phylogenetic Methods
● The methods used perform stochastic searches, that is, they do
not strive to maximize the likelihood, but rather integrate over it
● Thus, no numerical optimization methods for model parameters
and branch lengths are needed, parameters are proposed at
random
● It is substantially easier to infer trees under complex models
using Bayesian statistics than using Maximum Likelihood

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A Review of Probabilities
brown blonde Σ
light 5/40 15/40 20/40
dark 15/40 5/40 20/40
Σ 20/40 20/40 40/40
Hair color
Eye color

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brown blonde Σ
light 5/40 15/40 20/40
dark 15/40 5/40 20/40
Σ 20/40 20/40 40/40
Hair color
Eye color
Joint probability: probability of observing both A and B: Pr(A,B)
For instance, Pr(brown, light) = 5/40 = 0.125

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brown blonde Σ
light 5/40 15/40 20/40
dark 15/40 5/40 20/40
Σ 20/40 20/40 40/40
Hair color
Eye color
Marginal Probability: unconditional probability of an observation Pr(A)
For instance, Pr(dark) = Pr(dark,brown) + Pr(dark,blonde) = 15/40 + 5/40 = 20/40 = 0.5
Marginalize over hair color

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brown blonde Σ
light 5/40 15/40 20/40
dark 15/40 5/40 20/40
Σ 20/40 20/40 40/40
Hair color
Eye color
Conditional Probability: The probability of observing A given that B has occurred:
Pr(A|B) is the fraction of cases Pr(B) in which B occurs where A also occurs with Pr(AB)
Pr(A|B) = Pr(AB) / Pr(B)
For instance, Pr(blonde|light) = Pr(blonde,light) / Pr(light) = (15/40) / (20/40) = 0.75

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brown blonde Σ
light 5/40 15/40 20/40
dark 15/40 5/40 20/40
Σ 20/40 20/40 40/40
Hair color
Eye color
Statistical Independence: Two events A and B are independent
If their joint probability Pr(A,B) equals the product of their marginal probability Pr(A) Pr(B)
For instance, Pr(light,brown) ≠ Pr(light) Pr(brown), that is, the events are not independent!

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Joint Probability:
and
Bayes Theorem:
Pr(B|A) = Pr(A,B) / Pr(A)

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Joint Probability:
and
Bayes Theorem:
Pr(B|A) = Pr(A|B) Pr(B) / Pr(A)

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Bayes Theorem
Observed outcome
Unobserved outcome

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Bayes Theorem
Posterior probability
likelihood
Prior probability Marginal probability

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Bayes Theorem: Phylogenetics
Pr(Tree,Params|Alignment) = Pr(Alignment|Tree, Params) Pr(Tree,Params) / Pr(Alignment)
likelihood
Prior probability
Marginal probability
Posterior probability: distribution over all possible trees and all model parameter values
Likelihood: does the alignment fit the tree and model parameters?
Prior probability: introduces prior knowledge/assumptions about the probability distribution
of trees and model parameters (e.g., GTR rates, α shape parameter).
For instance, we typically assume that all possible tree topologies are equally probable
→ uniform prior
Marginal probability: how do we obtain this?

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Pr(Tree|Alignment) = Pr(Alignment|Tree) Pr(Tree) / Pr(Alignment)
likelihood
Prior probability
Marginal probability: Assume that our only model parameter is the tree and marginalizing
Means summing over all unconditional probabilities, thus
Pr(Alignment)
can be written as
Pr(Alignment) = Pr(Alignment, t0
) + Pr(Alignment,t1
) + … + Pr(Alignemnt, tn
)
where n+1 is the number of possible trees!

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likelihood
Prior probability
Marginal probability: Assume that our only model parameter is the tree and marginalizing
Means summing over all unconditional probabilities, thus
Pr(Alignment)
can be written as
Pr(Alignment) = Pr(Alignment, t0
) + Pr(Alignment,t1
) + … + Pr(Alignemnt, tn
)
where n+1 is the number of possible trees!
This can be re-written as
Pr(Alignment) = Pr(Alignment|t0
) Pr(t0
) + Pr(Alignment|t1
) Pr(t1
)+ … + Pr(Alignment|tn
) Pr(tn
)

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likelihood
Prior probability
Marginal probability:
Pr(Alignment) = Pr(Alignment|t0
) Pr(t0
) + Pr(Alignment|t1
) Pr(t1
)+ … + Pr(Alignment|tn
) Pr(tn
)
Now, we have all the ingredients for computing Pr(Tree|Alignment), however computing
Pr(Alignment) is prohibitive due to the large number of trees!
With continuous parameters the above equation for obtaining the marginal probability becomes
an integral. Usually, all parameters we integrate over (tree topology, model parameters, etc.) are
lumped into a parameter vector denoted by θ
likelihood
Prior := 1 / (n+1)
→ this is a uniform prior!

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Bayes Theorem General Form
f(θ|A) = f(A|θ) f(θ) / ∫f(θ)f(A|θ)dθ
Posterior distribution
likelihood
Prior distribution
Prior Probability
Marginal likelihood
Normalization constant
We know how to compute f(A|θ) → the likelihood of the tree
Problems:
Problem 1: f(θ) is given a priori, but how do we chose an appropriate distribution?
→ biggest strength and weakness of Bayesian approaches
Problem 2: How can we calculate/approximate ∫f(θ)f(A|θ)dθ ?
→ to explain this we need to introduce additional machinery
However, let us first look at an example for f(θ|A) in phylogenetics

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f(θ|A) = f(A|θ) f(θ) / ∫f(θ)f(A|θ)dθ
Note that, in the continuous case f() is called probability density function

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Probability Density Function
Properties:
1. f(x) > 0 for all allowed values x
2.The area under f(x) is 1.0
3.The probability that x falls into an interval (e.g. 0.2 – 0.3) is given by the
integral of f(x) over this interval

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An Example
1.0
Data (observations → sequences)
1.0
probability
Parameter space → 3 distinct tree topologies
Prior distribution
Posterior distribution
posterior
probability

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An Example
1.0
1.0
probability
Parameter space → 3 distinct tree topologies
Note that, this is a discrete
Distribution, since we only consider
the trees as parameters!
1/3 1/3 1/3
posterior
probability

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An Example
1.0
probability
What happens to the posterior
probability if we don't have enough data,
e.g., an alignment with a single site?
1/3 1/3 1/3
posterior
probability
?

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An Example
Parameter space of θ
Include additional model parameters such as branch lengths,GTR rates, and
the α-shape paremeter of the Г distribution into the model:
θ = (tree, α, branch-lengths, GTR-rates)
f(θ|A)
posterior
probability
Tree 1 tree 2 tree 3

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An Example
Marginal probability distribution of trees
We can look at this distribution for any parameter of interest by marginalizing
(integrating out) all other parameters.
Here we focus on the tree topology.
f(θ|A)
posterior
probability
20%
48%
32%

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An Example
Marginal probability distribution of trees
Here we focus on the tree topology.
f(θ|A)
posterior
probability
20%
48%
32%
We obtain the probability
by integrating over this
Interval!

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Marginalization
t1
t2
t3
α1
= 0.5 0.10 0.07 0.12 0.29
α2
= 1.0 0.05 0.22 0.06 0.33
α3
= 5.0 0.05 0.19 0.14 0.38
0.20 0.48 0.32 1.0
trees
Three discrete
Values of the
α-shape parameter
Joint probabilities
Marginal probabilities of trees
Marginal probabilities
of α values

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An Example
Marginal probability distibution of α
Here we focus on the three discrete α values.
f(θ|A)
posterior
probability
α = 5.0
29% 33% 38%
α = 1.0
α = 0.5

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Bayes versus Likelihood
ML: Joint estimation
Bayesian: Marginal estimation
See: Holder & Lewis
“Phylogeny Estimation: traditional &
Bayesian Approaches” Link to paper
likelihood

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Outline for today
● Monte-Carlo simulation & integration
● Markov-Chain Monte-Carlo methods

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f(θ|A) = (likelihood * prior) / ouch
Marginal likelihood
Normalization constant
→ difficult to calculate
We know how to compute f(A|θ) → the likelihood of the tree
Problems:
Problem 1: f(θ) is given a priori, but how do we chose an appropriate distribution
→ biggest strength and weakness of Bayesian approaches
Problem 2: How can we calculate/approximate ∫f(θ)f(A|θ)dθ
→ to explain this we need to introduce additional machinery to design methods for
numerical integration

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How can we compute this integral?
f(θ|A)

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The Classic Example
● Calculating π (the geometric constant!) with Monte-Carlo
Procedure:
1. Randomly throw points onto the
rectangle n times
2. Count how many points fall into
the circle ni
3. determine π as the ratio n / ni
→ this yields an approximation of
the ratio of the areas (the square
and the circle)

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Monte Carlo Integration
● Method for numerical integration of m-dimensional integrals over R:
∫f(θ)dθ ≈ 1/N Σ f(θi)
where θ is from domain Rm
●
More precisely, if the integral ∫ is defined over a domain/volume V
the equation becomes: V * 1/N * Σ f(θi)
● Key issues:
● Monte Carlo simulations draw samples θi of function f()
completely at random → random grid
● How many points do we need to sample for a 'good'
approximation?
● Domain Rm
might be too large for random sampling!

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Outline for today

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f(θ|A)
Monte-Carlo Methods: randomly sample data-points in this
huge parameter space to approximate the interval

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f(θ|A)
In which parts of the distribution are we interested?
Posterior
probability

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Distribution Landscape
f(θ|A)
Posterior
probability
Areas of high posterior
probability

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f(θ|A)
Posterior
probability
How can we get a
sample faster?

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f(θ|A)
Posterior
probability
How can we get a
sample faster? →
Markov Chain Monte Carlo
Methods

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Distribution Landcsape
f(θ|A)
Posterior
probability
Higher sample density Higher sample density

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Distribution Landcsape
f(θ|A)
Posterior
probability
Fewer misses

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Markov-Chain Monte-Carlo
f(θ|A)
Posterior
probability
MCMC → biased random walks: the probability to evaluate/find a sample in an area
with high posterior probability is proportional to the posterior distribution

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Markov-Chain Monte-Carlo
● Idea: Move the grid/samples into regions of high probability
● Construct a Markov Chain that generates samples such that
more time is spent (more samples are evaluated) in the most
interesting regions of the state space
● MCMC can also be used for hard CS optimization problems, for
instance, the knapsack problem
● Note that, MCMC is similar to Simulated Annealing → there's no
time to go into the details though here!

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The Robot Metaphor
Crete, December 2019

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The Robot Metaphor
● Drop a robot onto an unknown planet to explore its landscape
● Teaching idea and slides adapted from Paul O. Lewis
Uphill steps → always accepted
Small downhill steps
→ usually accepted Huge downhill steps
→ almost never accepted
elevation

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How to accept/reject proposals
● Decision to accept/reject a proposal to go from
Point 1 → Point 2 is based on the ratio R of posterior densities
of the two points/samples
R = Pr(Point2|data) / Pr(point1|data) =
(Pr(Point2)Pr(data|point2) / Pr(data)) / (Pr(Point1)Pr(data|point1) / Pr(data))
= Pr(point2)Pr(data|point2) / Pr(point1)Pr(data|point1)

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(Pr(Point2)Pr(data|point2) / Pr(data)) / (Pr(Point1)Pr(data|point1) / Pr(data))
= Pr(point2)Pr(data|point2) / Pr(point1)Pr(data|point1)
The marginal probability of the data cancels out!
Phew, we don't need to compute it.

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(Pr(Point2)Pr(data|point2) / Pr(data)) / (Pr(Point1)Pr(data|point1) / Pr(data)) =
(Pr(point2)/Pr(point1)) * (Pr(data|point2)Pr(data|point1))
Prior ratio: for uniform priors this is 1 !
Likelihood ratio

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The Robot Metaphor
● Drop a robot onto an unknown planet to explore its landscape
At 1m, proposed to go to 2
Ratio = 2/1 → accept
At 10 m, go down to 9 m
Ratio: 9/10 = 0.9 → accept with
probability 90%
At 8 m, go down to 1m
Ratio: 1/8 = 0.125 → accept
with probability 12.5%
elevation

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Distributions
● The target distribution is the posterior distribution we are trying
to sample (integrate over)!
● The proposal distribution decides which point (how far/close) in
the landscape to randomly go to/try next:
→ The choice has an effect on the efficiency of the MCMC
algorithm, that is, how fast it will get to these interesting areas we
want to sample

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The Robot Metaphor
Target distribution/
posterior probability
Proposal distribution: how far
Left or right will we usually go?

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The Robot Metaphor
Proposal distribution:
with smaller variance →
what happens?
Pros: Seldom refuses a step
Cons: smaller steps, more steps
required for exploration

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The Robot Metaphor
with larger variance →
What happens?

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The Robot Metaphor
with larger variance →
what happens?
Pros: can cover a large area
quickly
Cons: lots of steps will be rejected

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The Robot Metaphor
A proposal distribution that
balances pros & cons yields
'good mixing'

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Mixing
● A well-designed chain will require a few steps until reaching
convergence, that is, approximating the underlying probability
density function 'well-enough' from a random starting point
● It is a somewhat fuzzy term, refers to the proportion of accepted
proposals (acceptance ratio) generated by a proposal mechanism
→ should be neither too low, nor too high
● The real art in designing MCMC methods consists
● building & tuning good proposal mechanisms
● selecting appropriate proposal distributions
● such that they quickly approximate the distribution we want to sample
from

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The Robot Metaphor
When the proposal distribution is
symmetric, that is, the probability
of moving left or right is the same,
we use the Metropolis algorithm

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The Metropolis Algorithm
● Metropolis et al. 1953 http://www.aliquote.org/pub/metropolis-et-al-1953.pdf
● Initialization: Choose an arbitrary point θ0 as first sample
● Choose an arbitrary probability density Q(θi+1|θi ) which suggests a candidate for the next
sample θi+1 given the previous sample θi.
● For the Metropolis algorithm, Q() must be symmetric:
it must satisfy Q(θi+1|θi ) = Q(θi|θi+1)
● For each iteration i:
● Generate a candidate θ* for the next sample by picking from the distribution Q(θ*|θi )
● Calculate the acceptance ratio R = Pr(θ*)Pr(data|θ*) / Pr(θi )Pr(data/θi )
– If R ≥ 1, then θ* is more likely than θi → automatically accept the candidate by setting θi+1 :=
θ*
– Otherwise, accept the candidate θ* with probability R → if the candidate is rejected: θi+1 := θi

64
The Metropolis Algorithm
● Metropolis et al. 1953 http://www.aliquote.org/pub/metropolis-et-al-1953.pdf
● Initialization: Choose an arbitrary point θ0 as first sample
● Choose an arbitrary probability density Q(θi+1|θi ) which suggests a candidate for the next
sample θi+1 given the previous sample θi.
● For the Metropolis algorithm, Q() must be symmetric:
it must satisfy Q(θi+1|θi ) = Q(θi|θi+1)
● Generate a candidate θ* for the next sample by picking from the distribution Q(θ*|θi )
● Calculate the acceptance ratio R = Pr(θ*)Pr(data|θ*) / Pr(θi )Pr(data/θi )
– If R ≥ 1, then θ* is more likely than θi → automatically accept the candidate by setting θi+1 :=
θ*
– Otherwise, accept the candidate θ* with probability R → if the candidate is rejected: θi+1 := θi
Conceptually this is the same Q
we saw for substitution models!

65
Phylogenetic Metropolis Algorithm
● Initialization: Choose a random tree with random branch lengths as first sample
● Propose either
– a new tree topology
– a new branch length
and re-calculate the likelihood
● Calculate the acceptance ratio of the proposal
● Accept the new tree/branch length or reject it
● Print current tree with branch lengths to file only every k (e.g. 1000) iterations
→ to generate a sample from the chain
→ to avoid writing TBs of files
→ also known as thinning
● Summarize the sample using means, histograms, credible intervals, consensus trees,
etc.

66
Uncorrected Proposal Distribution
A Robot in 3D
Example: MCMC proposed moves to
the right 80% of the time without Hastings
correction for acceptance probability!
Peak area

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Hastings Correction
We need to decrease chances to
move to the right by 0.5 and
Increase chances to move to the
left by factor 2 to compensate for
the asymmetry!
1/3 2/3

68
Hastings Correction
R = (Pr(point2)/Pr(point1)) * (Pr(data|point2)/Pr(data|point1)) * (Q(point1|point2) / Q(point2|point1))
Prior ratio: for uniform priors this is 1 !
Likelihood ratio
Hastings ratio: if Q is symmetric
Q(point1|point2) = Q(point2|point) and
the hastings ratio is 1 → we obtain the
normal Metropolis algorithm

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Hastings Correction
more formally
R = (f(θ*)/f(θi )) * (f(data|θ*)/f(data|θi )) * (Q(θi |θ*) / Q(θ*|θi ))
Prior ratio
Likelihood ratio
Hastings ratio

70
Hastings Correction is not trivial
● Problem with the equation for the hastings correction
● M. Holder, P. Lewis, D. Swofford, B. Larget. 2005.
Hastings Ratio of the LOCAL Proposal Used in Bayesian
Phylogenetics. Systematic Biology. 54:961-965.
http://sysbio.oxfordjournals.org/content/54/6/961.full
“As part of another study, we estimated the marginal likelihoods
of trees using different proposal algorithms and discovered
repeatable discrepancies that implied that the published
Hastings ratio for a proposal mechanism used in many
Bayesian phylogenetic analyses is incorrect.”
● Incorrect Hastings ratio used from 1999-2005

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Back to Phylogenetics
A
B
C
D
E
A
B
C
D
E
A
C
D
A
E
D
C
B
A
B
C
E
D
A
C
E
D
B
A
B
D
C
E
What's the posterior probability of bipartition AB|CDE?

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Back to Phylogenetics
A
B
C
D
E
A
B
C
D
E
A
C
D
A
E
D
C
B
A
B
C
E
D
A
C
E
D
B
A
B
D
C
E
What's the posterior probability of bipartition AB|CDE?
We just count from the sample generated by MCMC, here it's 3/5 → 0.6
This approximates the true proportion (posterior probability) of bipartition AB|CDE
if we have run the chain long enough and if it has converged

73
MCMC in practice
Frequency of AB|CDE
generations
convergence
Burn-in → discarded from our final sample
Random
starting point

74
Convergence
● How many samples do we need to draw to obtain an accurate
approximation?
● When can we stop drawing samples?
● Methods for convergence diagnosis
→ we can never say that a MCMC-chain has converged
→ we can only diagnose that it has not converged
→ a plethora of tools for convergence diagnostics for
phylogenetic MCMC

75
Convergence
Entire landscape
Likelihood
score
Likelihood
Score output
MCMC method
Area of apparent
convergence
Zoom in

76
Solution: Run Multiple Chains
Robot 1
Robot 2

77
Outline for today

78
Heated versus Cold Chains
Robot 1
Robot 2
Cold chain: sees
landscape as is
Hot chain: sees a
Flatter version of the
same landscape →
Moves more easily
between peaks

79
Known as MCMCMC
● Metropolis-Coupled Markov-Chain Monte Carlo
● Run several chains simultaneously
● 1 cold chain (the one that samples)
● Several heated chains
● Heated chain robots explore the parameter space in larger
steps
● To flatten the landscape the acceptance ratio R is modified as
follows: R1/1+H where H is the so-called temperature
– For the cold chain H := 0.0
– Setting the temperature for the hot chains is a bit of woo-
do

80
Robot 1: cold
Robot 2: hot
Exchange information every now and then

81
Robot 1: hot
Robot 2: cold
Swap cold/hot states to better sample
this nice peak here

82
Robot 1: hot
Robot 2: cold
Decision on when to swap is a bit more
complicated!

83
Robot 1: hot
Robot 2: cold
Only the cold robot actually emits states (writes samples to file)

84
A few words about priors
● Prior probabilities convey the scientist's beliefs, before having
seen the data
● Using uninformative prior probability distributions (e.g., uniform
priors, also called flat priors)
→ differences between prior and posterior distribution are
attributable to likelihood differences
● Priors can bias an analysis
● For instance, we could chose an arbitrary prior distribution for
branch lengths in the range [1.0,20.0]
→ what happens if branch lengths are much shorter?

85
Some Phylogenetic Proposal
Mechanisms
● Branch Lengths
● Sliding Window Proposal
● Multiplier Proposal
● Topologies
● Local Proposal (the one with the bug in the Hastings ratio)
● Extending TBR (Tree Bisection Reconnection) Proposal
● Remember: We need to design proposals for which
● We either don't need to calculate the Hastings ratio
● Or for which we can calculate it
● That have a 'good' acceptance rate
→ all sorts of tricks being used, e.g., parsimony-biased topological proposals

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