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Character # Taxon 1 2 3 4 5 6 7 8 9 10 A 0 0 0 0 0 - - PowerPoint PPT Presentation

Jan 10, 2023 140 likes •503 views

Character # Taxon 1 2 3 4 5 6 7 8 9 10 A 0 0 0 0 0 0 0 0 0 0 B 1 0 0 0 0 1 1 1 1 1 C 0 1 1 1 0 1 1 1 1 1 D 0 0 0 0 1 1 1 1 1 0 B 1 B C D 10 A C 6

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SLIDE 1

Character # Taxon 1 2 3 4 5 6 7 8 9 10 A B 1 1 1 1 1 1 C 1 1 1 1 1 1 1 1 D 1 1 1 1 1

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SLIDE 2

B C D A B C 1 2 3 4 5 10 6 7 8 9

✬ ✫ ✩ ✪

D

✬ ✫ ✩ ✪

A

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SLIDE 3

Interestingly, without polarization Hennig’s method can infer unrooted

  • trees. We can get the tree topology, but be unable to tell paraphyletic from

monophyletic groups. The outgroup method amounts to inferring an unrooted tree and then rooting the tree on the branch that leads to an outgroup.

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SLIDE 4

B C D A B A C D 1 2 3 4 5 10 6 7 8 9

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SLIDE 5

Inadequacy of logic

Unfortunately, though Hennigian logic is valid we quickly find that we do not have a reliable method of generating accurate homology statements. The logic is valid, but we don’t know that the premises are true. In fact, we almost always find that it is impossible for all of our premises to be true.

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SLIDE 6

Character conflict

Homo sapiens AGTTCAAGT Rana catesbiana AATTCAAGT Drosophila melanogaster AGTTCAAGC

  • C. elegans

AATTCAAGC The red character implies that either (Homo + Drosophila) is a group (if G is derived) and/or (Rana + C. elegans) is a group. The green character implies that either (Homo + Rana) is a group (if T is derived) and/or (Drosophila + C. elegans) is a group. The green and red character cannot both be correct.

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SLIDE 7

Character # Taxon 1 2 3 4 5 6 7 8 9 10 11 12 A B 1 1 1 1 1 1 1 1 C 1 1 1 1 1 1 1 1 1 D 1 1 1 1 1 1

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SLIDE 8

C B D

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

A

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SLIDE 9

Character conflict

Two characters are compatible if they can both be mapped on the same tree so that all of the character states displayed could be homologous. Incompatible characters are evidence of homoplasy in the data Homoplasy literally means the “same change” has occurred more than once in the evolutionary history of the group. The presence of homoplasy undermines Hennigian analyses.

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SLIDE 10

white = space

  • f all possible

matrices

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SLIDE 11

blue = space

  • f matrices with

the pattern: A B C D

  • * * -
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SLIDE 12

red = space

  • f matrices with

the pattern: A B C D

  • * - *
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SLIDE 13

yellow = space

  • f matrices with

the pattern: A B C D

  • - * *
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SLIDE 14

all eight categories of matrices

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SLIDE 15

blue = space

  • f matrices

compatible with tree: (A,(B,C),D)

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SLIDE 16

blue = space

  • f matrices

compatible with tree: (A,C,(B,D))

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SLIDE 17

blue = space

  • f matrices

compatible with tree: (A,B,(C,D))

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SLIDE 18

Hennigian: grey = any tree blue = B+C red = B+D yellow = C+D white = no tree (conflicting characters)

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SLIDE 19

A B C D

A 0000000000 B 1111111111 C 1111111111 D 1111111111 A 0000000000 B 1111111110 C 1111111111 D 1111111111 A 0000000000 B 1111111111 C 1111111110 D 1111111111 A 0000000000 B 1111111111 C 1111111111 D 1111111110 A 0000000000 B 1111111110 C 1111111111 D 1111111110 A 0000000000 B 1111111111 C 1111111110 D 1111111110 A 0000000000 B 1111111101 C 1111111111 D 1111111111 A 0000000000 B 1111111100 C 1111111111 D 1111111111 A 0000000000 B 1111111101 C 1111111110 D 1111111111 A 0000000000 B 1111111110 C 1111111110 D 1111111111

A D C B A C B D

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SLIDE 20

What can we do if our data end up in the white (character conflict) or grey (uninformative characters only) zone?

  • can we detect character conflict?
  • is there a logic-based solution to the problem of character conflict?
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SLIDE 21

Detecting character conflict in binary characters

Consider the four possible combinations of states in a two-character matrix. The characters are incompatible iff (when you look across all taxa) you see all four state combinations. Char 1 1 Char 2 × × 1 × ×

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SLIDE 22

What can we do if our data end up in the white (character conflict) or grey (uninformative characters only) zone?

  • Can we detect character conflict? Yes
  • Is there a logic-based solution to the problem of character conflict?

– recoding characters? – “reciprocal illumination”?

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SLIDE 23

What can we do if our data end up in the white (character conflict) or grey (uninformative characters only) zone?

  • Can we detect character conflict? Yes
  • Is there a logic-based solution to the problem of character conflict? No,

nothing purely based on logic (and the suggestions for culling data to make matrices suitable for logical inference can lead to unsatisfyingly subjecive analyses).

  • What can we do?

We must have an “error model”

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SLIDE 24

Statistical inference

There are many ways to derive estimators, we are going to talk about maximum likelihood estimation:

θ ∈ Θ X ∈ X x ∼ Pr(X = x|θ) L(θ) = Pr(X = x|θ) ˆ θ = arg max L(θ)

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SLIDE 25

A B C D A D B C A C B D

Θ

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SLIDE 26

A B C D

θ ∈ Θ

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SLIDE 27

A 0000000000 B 1111111111 C 1111111111 D 1111111111 A 0000000000 B 1111111110 C 1111111111 D 1111111111 A 0000000000 B 1111111111 C 1111111110 D 1111111111 A 0000000000 B 1111111110 C 1111111110 D 1111111111 A 0000000000 B 1111111111 C 1111111111 D 1111111110 A 0000000000 B 1111111110 C 1111111111 D 1111111110 A 0000000000 B 1111111111 C 1111111110 D 1111111110 A 0000000000 B 1111111101 C 1111111111 D 1111111111 A 0000000000 B 1111111100 C 1111111111 D 1111111111 A 0000000000 B 1111111101 C 1111111110 D 1111111111

...

X

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SLIDE 28

A B C D

A 0000000000 B 1111111111 C 1111111111 D 1111111111 A 0000000000 B 1111111110 C 1111111111 D 1111111111 A 0000000000 B 1111111111 C 1111111110 D 1111111111 A 0000000000 B 1111111110 C 1111111110 D 1111111111 A 0000000000 B 1111111111 C 1111111111 D 1111111110 A 0000000000 B 1111111110 C 1111111111 D 1111111110 A 0000000000 B 1111111111 C 1111111110 D 1111111110 A 0000000000 B 1111111101 C 1111111111 D 1111111111 A 0000000000 B 1111111100 C 1111111111 D 1111111111 A 0000000000 B 1111111101 C 1111111110 D 1111111111

0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024

Pr(X = x|θ)

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SLIDE 29

A B C D

A 0000000000 B 1111111111 C 1111111111 D 1111111111 A 0000000000 B 1111111110 C 1111111111 D 1111111111 A 0000000000 B 1111111111 C 1111111110 D 1111111111 A 0000000000 B 1111111110 C 1111111110 D 1111111111 A 0000000000 B 1111111111 C 1111111111 D 1111111110 A 0000000000 B 1111111110 C 1111111111 D 1111111110 A 0000000000 B 1111111111 C 1111111110 D 1111111110 A 0000000000 B 1111111101 C 1111111111 D 1111111111 A 0000000000 B 1111111100 C 1111111111 D 1111111111 A 0000000000 B 1111111101 C 1111111110 D 1111111111

x ∼ Pr(X = x|θ)

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SLIDE 30

A 0000000000 B 1111111110 C 1111111110 D 1111111111

x represents

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SLIDE 31

A B C D A D B C A C B D

A 0000000000 B 1111111110 C 1111111110 D 1111111111

? ? ?

θ1 θ2 θ3

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SLIDE 32

A B C D A D B C A C B D

A 0000000000 B 1111111110 C 1111111110 D 1111111111

θ1 θ2 θ3 Pr(x|θ1) = 0.00024

0.00024

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SLIDE 33

A B C D A D B C A C B D

A 0000000000 B 1111111110 C 1111111110 D 1111111111

θ1 θ2 θ3 Pr(x|θ2) = 0.0002

0.00024 0.0002

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SLIDE 34

A B C D A D B C A C B D

A 0000000000 B 1111111110 C 1111111110 D 1111111111

θ1 θ2 θ3 Pr(x|θ3) = 0.00022

0.00024 0.0002 0.00022

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SLIDE 35

A B C D A D B C A C B D

A 0000000000 B 1111111110 C 1111111110 D 1111111111

ˆ θ = arg max L(θ)

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SLIDE 36

ML Estimation

  • Flexible form of inference
  • Requires a model: Pr(X = x|θ)

Under mild conditions, ML estimation is asymptotically:

  • not very biased,
  • efficient

How can we come up with a model?