An Equivalence of Maximum Parsimony and Maximum Likelihood revisited

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

2. 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

3. 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

4. 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

5. Linearity of the Likelihood Function • An extensiongf 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! c1 c2 c1 c2 1 2 3 4 f: c1 c1 c1 c2 Mareike Fischer

6. 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

7. 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

8. Bound of the Likelihood Function Let k be the number of ∞-edges. 0 As before, we have ∞ ∞ ∞ 0 Note that P(gf)=0 if gfrequires a substitution on an edge of length 0! Therefore, 0 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

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

10. Equivalence of MP and ML • So we have: • But obviously also • as the most parsimonious extension of f requires exactly PS(f,T) changes. Altogether: 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

11. 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

12. Example: Bounded edge lengths for r=2 Then, PS(f1|T1) = PS(f2|T2) = 1 Therefore, MP and ML are notequivalent in this setting! Also, P(f1|T1) = P(f2|T2),  MP is indecisive between T1 and T2 ! but max P(f2|T1) = 2u2(1-u)2 > u2 = max P(f1|T2) Note that by repeating f1n times and f2(n+c) times (c>0), a strongcounterexample can be constructed!  ML favors T1 over T2 ! and PS(f1|T2) = PS(f2|T1) = 2 Mareike Fischer

13. Example: Molecular clock 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

14. 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

15. 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