Mareike Fischer

An Equivalence of Maximum Parsimony andMaximum Likelihood revisited

Mareike Fischer

and Bhalchandra Thatte

MIEP, 10 – 12 June 08, Montpellier

Mareike Fischer

The Problem

• Growing amount of DNA data

stochastic models and methods needed for analysis!

• MP and ML are two of the most frequently discussed methods.

• MP and ML can perform differently (e.g. in the so-called ‘Felsenstein Zone’)

• But: When are MP and ML equivalent?

Approach by Tuffley & Steel

Mareike Fischer

• Given: r character states c1,…,cr ;

• No distinction between character states (fully

symmetric model!);

• The probability pe of a transition on edge e is

pe ≤ 1/r;

• Transition events on different edges are independent.

Note: If r=4: Jukes-Cantor!

The Nr-Model

Mareike Fischer

Tuffley and Steel (1997):

MP and ML with no common mechanism are equivalent in the sense that both choose the same tree(s).

Note: ‘No common mechanism’ means that the transition probabilities can vary from site to site.

The Equivalence Result

Mareike Fischer

An extension gf of a character f agrees with f on the leaves, but also assigns character states to the ancestral nodes.

Example: r=2, f=(c1,c1,c1,c2):

Linearity of the Likelihood Function

f: c1 c1 c1 c2

1 2 3 4

c1 c1

c1

c2

c2

c2

8 different extensions!

Mareike Fischer

Note that

and

Linearity of the Likelihood Function

c1

u

1 2 3 4

pe

Thus, P(f) is linear in each pe !

Mareike Fischer

Maximum of the Likelihood Function

Linear functions h: [0,t] kR are maximized at a corner of the box [0,t] k.

t

tThus, we can assume wlog. that ML chooses a tree T with pe = 0 or 1/r for all edges e of T !

1/r

1/r

Mareike Fischer

Bound of the Likelihood Function

0

∞ ∞

∞

∞

0

As before, we have

Note that P(gf)=0 if gf

requires a substitution on an edge of length 0!

Note that if P(gf)≠0 , then P(gf)=(1/r) k+1 !

Let k be the number of ∞-edges.

ML-Tree T

Therefore,

For N = #{gf : P(gf)≠0}

And thus

00

Mareike Fischer

Bound for the Likelihood Function

0

∞ ∞

∞

∞

000

So, for N = #{gf : P(gf)≠0} and k = #{∞-edges}, we have:

Wanted: Upper bound for N .

c1 ci

cj

ckck

• Delete ∞-edges;

• k+1 connected components remain,

• M of them are labelled (i.e. contain at least one leaf)

Here: k =4.

k+1 components,

M labelled

And: PS(f,T) ≤ M – 1

Mareike Fischer

So we have:

But obviously also

as the most parsimonious extension of f requires exactly PS(f,T) changes.

Equivalence of MP and ML

Altogether:

And thus

Applied to one character f, MP and ML are equivalent!

In a sequence of ‘no common mechanism’, each likelihood can be maximized independently, and thus

Mareike Fischer

Bounded edge lengths

Modification of the model: Transition probabilities subject to upper bound u:

0 ≤ pe ≤ u < 1/r

Then, MP and ML are not equivalent!

Mareike Fischer

Example: Bounded edge lengths for r=2

Then, PS(f1|T1) = PS(f2|T2) = 1

and PS(f1|T2) = PS(f2|T1) = 2

MP is indecisive between T1 and T2 !

Also, P(f1|T1) = P(f2|T2),but

max P(f2|T1) = 2u2(1-u)2 > u2 = max P(f1|T2)

ML favors T1 over T2 !

Therefore, MP and ML are not equivalent in this setting!

Note that by repeating f1 n times and f2 (n+c) times (c>0), a strong counterexample can be constructed!

Mareike Fischer

Molecular clock

Under a molecular clock, MP and ML are not equivalent!

Note that under a clock, the maximum of the likelihood can occur in the interior of the box

[0,1/r]k !

Here, pe = (1-Pe)/2.

Example:

The ‘height’ P of the tree is fixed: P=P1P2=P3P4P5

In this setting, MP is indecisive between T1

and T2 but ML favors T1.

Mareike Fischer

Summary

Even under the assumption of no common mechanism, MP and ML do not have to be equivalent! Small changes to the model assumptions suffice to achieve this.

Mareike Fischer

Thanks…

… to my supervisor Mike Steel,

… to the organizers of this conference,

… to the Allan Wilson Centre for financing my research,

… to YOU for listening or at least waking up early enough to read this message .