Climbing Peaks and Crossing Valleys: Metropolis Coupling and Rugged Phylogenetic Distributions

Climbing Peaks and Crossing Valleys: Metropolis Coupling and Rugged Phylogenetic Distributions

Jeremy M. Brown Robert C. Thomson

@jembrown www.phyleauxgenetics.org

Bayesian Inference Requires Integration

Tree,Parameter Space

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Ƭ1

Markov Chain Monte Carlo (MCMC)

Tree,Parameter Space

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1) Start somewhere 2) Propose a new position 3) Calculate posterior density

ratio (r) of new to old states - If r > 1, accept - If r < 1, accept with

probability r. 4) Record state. 5) Repeat many times.

Yes!Maybe

Markov Chain Monte Carlo (MCMC)

Tree,Parameter Space

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MCMC Has Trouble With Rugged Distributions

Tree,Parameter Space

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Ƭ2

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Tree,Parameter Space

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Tree,Parameter Space

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MCMC Has Trouble With Rugged Distributions

Tree,Parameter Space

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MCMC Has Trouble With Rugged Distributions

Tree,Parameter Space

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Bipartition Bayes Factors

A

B

C

E

D

Marginal likelihood with AB | CDE

Bayes Factor

Marginal likelihood without AB | CDE + -

Negative Constraints = Rugged Distributions

Negative Constraints = Rugged Distributions

homo_sapiens

pantherophis_guttata

zebra_finch

anolis_carolinensis

gallus_gallus

alligator_mississippiensis

crocodylus_porosus

pelomedusa_subrufa

sphenodon_tuatara

chrysemys_picta

homo_sapiens

chrysemys_picta

sphenodon_tuatara

zebra_finch

anolis_carolinensis

gallus_gallus

alligator_mississippiensis

pantherophis_guttata

pelomedusa_subrufa

crocodylus_porosus

zebra_finchhomo_sapiens

crocodylus_porosus

sphenodon_tuatara

pantherophis_guttata

chrysemys_picta

alligator_mississippiensis

gallus_gallus

anolis_carolinensis

pelomedusa_subrufa

Alternative Insertion Swaps are Difficult

homo_sapiens

pantherophis_guttata

zebra_finch

anolis_carolinensis

gallus_gallus

alligator_mississippiensis

crocodylus_porosus

pelomedusa_subrufa

sphenodon_tuatara

chrysemys_picta zebra_finchhomo_sapiens

crocodylus_porosus

sphenodon_tuatara

pantherophis_guttata

chrysemys_picta

alligator_mississippiensis

gallus_gallus

anolis_carolinensis

pelomedusa_subrufa

Data

Data

The Po-Boy Problem

How do you change the seafood on your po-boy while someone’s holding the sandwich?

Shrimp

Oysters

Halves of french roll = Naturally monophyletic taxa

Seafood = Inserted taxon

Metropolis Coupling (MC3) Improves Mixing

Tree,Parameter Space

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can act as “scouts”.

Swap?

Peaks All Found, But Different Probabilities?

homo_sapiens

chrysemys_picta

sphenodon_tuatara

zebra_finch

anolis_carolinensis

gallus_gallus

alligator_mississippiensis

pantherophis_guttata

pelomedusa_subrufa

crocodylus_porosus

homo_sapiens

pantherophis_guttata

zebra_finch

anolis_carolinensis

gallus_gallus

alligator_mississippiensis

crocodylus_porosus

pelomedusa_subrufa

sphenodon_tuatara

chrysemys_pictazebra_finchhomo_sapiens

crocodylus_porosus

sphenodon_tuatara

pantherophis_guttata

chrysemys_picta

alligator_mississippiensis

gallus_gallus

anolis_carolinensis

pelomedusa_subrufa0.500.25

0.240.38

0.250.24

Run 1Run 2

GenerationLn

L

A Closer Look at the Acceptance Ratio

r =pi(⌧j , ✓j |D) pj(⌧i, ✓i|D)

pi(⌧i, ✓i|D) pj(⌧j , ✓j |D)

A Closer Look at the Acceptance Ratio

Does chain i like where chain j is?

Does chain j like where chain i is?

r =pi(⌧j , ✓j |D) pj(⌧i, ✓i|D)

pi(⌧i, ✓i|D) pj(⌧j , ✓j |D)

A Closer Look at the Acceptance Ratio

r =pi(⌧j , ✓j |D) pj(⌧i, ✓i|D)

pi(⌧i, ✓i|D) pj(⌧j , ✓j |D)

r =

p(⌧j , ✓j |D)

p(⌧i, ✓i|D)

� 1Ti

� 1Tj

A Closer Look at the Acceptance Ratio

r =pi(⌧j , ✓j |D) pj(⌧i, ✓i|D)

pi(⌧i, ✓i|D) pj(⌧j , ✓j |D)

r =

p(⌧j , ✓j |D)

p(⌧i, ✓i|D)

� 1Ti

� 1Tj

When temps equal, ALL swaps accepted regardless of posterior density.

A Simple One-Parameter Example

0.0 0.2 0.4 0.6 0.8 1.0

01

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Parameter Value

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0.2

https://github.com/jembrown/toyMC3/

Max Temp > Number of Chains

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Maximum Temperature

Peak O

ne P

robability

5 Chains

10 Chains

20 Chains

0.0 0.2 0.4 0.6 0.8 1.0

01

23

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Parameter Value

Pro

bability D

ensity

0.8

0.2

Peaks Have Different “Capture” Probabilities

0.0 0.2 0.4 0.6 0.8 1.0

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Parameter Value

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P=0.8 P=0.2

Spurious Convergence by Chain Number

0.0 0.2 0.4 0.6 0.8 1.0

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Parameter Value

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0.8

0.2

P=0.8 P=0.2

When two runs end up with the same distribution

of poorly mixing chains across peaks,

they will estimate nearly identical (but incorrect!)

probabilities.

Lots of Chains Looks Like Convergence

2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

Maximum Temperature

Peak O

ne P

robability/S

tandard

Devia

tion

5 Chains

10 Chains

20 Chains

0.0 0.2 0.4 0.6 0.8 1.0

01

23

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Parameter Value

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en

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0.8

0.2

0.0 0.2 0.4 0.6 0.8 1.0

01

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Parameter Value

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0.8

0.2

Peak One0.8 * N

Peak Two0.2 * N

P=0.8 P=0.2

N (large #) Chains

Law of Large Numbers

Lots of Chains Looks Like Convergence

Negative Constraint on Bird Monophyly

zebra_finchhomo_sapiens

crocodylus_porosus

sphenodon_tuatara

pantherophis_guttata

chrysemys_picta

alligator_mississippiensis

gallus_gallus

anolis_carolinensis

pelomedusa_subrufa

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Maximum Temperature

Pro

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2 Chains4 Chains8 Chains16 Chains32 Chains

Negative Constraint on Bird Monophyly

zebra_finchhomo_sapiens

crocodylus_porosus

sphenodon_tuatara

pantherophis_guttata

chrysemys_picta

alligator_mississippiensis

gallus_gallus

anolis_carolinensis

pelomedusa_subrufa

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Maximum Temperature

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Sta

ndar

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tion 2 Chains

4 Chains8 Chains16 Chains32 Chains

Warnings

• Despite improving mixing, MC3 analyses still require careful thought.

• With small numbers of chains and small numbers of runs, estimated probabilities can be incorrect but identical across some runs.

• With large numbers of chains, estimated probabilities become increasingly similar across all runs.

Broad v Rugged Distributions

Tree,Parameter Space

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Recommendations

• For rugged distributions, increase maximum chain temperature not chain number

• For broad distributions, increase chain number

• Use more than 2 runs

Thank You

DEB-1355071DEB-1354506

@jembrown

Michael Landis

Karen Cranston

Negative Constraints = Rugged Distributions

TreeScaper

Guifang Zhou (SSB symposium lightning talk) - Monday, 1:45-1:50 - Ballroom A "A network framework to explore phylogenetic structure in genome data"

Guifang Zhou (iEvoBio talk) - Tuesday, 2:05-2:12 - Meeting Room 9C"TreeScaper: Software to visualize and extract phylogenetic signals from sets of trees”

https://github.com/whuang08/TreeScaper

Spurious Convergence by Chain Number

0.0 0.2 0.4 0.6 0.8 1.0

01

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Parameter Value

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0.8

0.2

2 Chains, 0 Chains0.64

1 Chain, 1 Chain0.32

0 Chains, 2 Chains0.04 P=0.8 P=0.2

2 Chains