An Equivalence of Maximum Parsimony and Maximum Likelihood revisited MIEP, 10 – 12 June 08, Montpellier An Equivalence of Maximum Parsimony and Maximum Likelihood revisited Mareike Fischer and Bhalchandra Thatte 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

The Nr-Model 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! Mareike Fischer

The Equivalence Result 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. Mareike Fischer

Linearity of the Likelihood Function 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): c1 c2 8 different extensions! c2 c1 c1 c2 1 2 3 4 f: c1 c1 c1 c2 Mareike Fischer

Thus, P(f) is linear in each pe ! Linearity of the Likelihood Function Note that and u pe Thus, P(f) is linear in each pe ! 1 2 3 4 c1 Mareike Fischer

Maximum of the Likelihood Function Linear functions h: [0,t] kR are maximized at a corner of the box [0,t] k. Thus, we can assume wlog. that ML chooses a tree T with pe = 0 or 1/r for all edges e of T ! 1/r t t 1/r Mareike Fischer

Bound of the Likelihood Function Let k be the number of ∞-edges. As before, we have ∞ ∞ ∞ Therefore, Note that P(gf)=0 if gf requires a substitution on an edge of length 0! For N = #{gf : P(gf)≠0} ML-Tree T ∞ Note that if P(gf)≠0 , then P(gf)=(1/r) k+1 ! And thus Mareike Fischer

So, for N = #{gf : P(gf)≠0} and k = #{∞-edges}, we have: Bound for the Likelihood Function So, for N = #{gf : P(gf)≠0} and k = #{∞-edges}, we have: ∞ Wanted: Upper bound for N . ∞ ∞ Delete ∞-edges; k+1 connected components remain, M of them are labelled (i.e. contain at least one leaf) And: PS(f,T) ≤ M – 1 ck ck c1 ci k+1 components, M labelled ∞ Here: k =4. cj Mareike Fischer

Equivalence of MP and ML Altogether: So we have: But obviously also as the most parsimonious extension of f requires exactly PS(f,T) changes. And thus In a sequence of ‘no common mechanism’, each likelihood can be maximized independently, and thus  Applied to one character f, MP and ML are equivalent! Mareike Fischer

Then, MP and ML are not equivalent! 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 Therefore, MP and ML are not equivalent in this setting! Also, P(f1|T1) = P(f2|T2),  MP is indecisive between T1 and T2 ! Note that by repeating f1 n times and f2 (n+c) times (c>0), a strong counterexample can be constructed! but max P(f2|T1) = 2u2(1-u)2 > u2 = max P(f1|T2)  ML favors T1 over T2 ! and PS(f1|T2) = PS(f2|T1) = 2 Mareike Fischer

Under a molecular clock, MP and ML are not equivalent! Example: Here, pe = (1-Pe)/2. 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 ! 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 . Mareike Fischer