Computing likelihoods under Λ-coalescents

Transcription

1 Computing likelihoods under Λ-coalescents Matthias Birkner LMU Munich, Dept. Biology II Joint work with Jochen Blath and Matthias Steinrücken, TU Berlin New Mathematical Challenges from Molecular Biology, Banff International Research Station, September 6-11, 2009 Matthias Birkner (LMU Munich) Banff, Sept / 42

2 Outline 1 Introduction 2 Infinitely-many-sites model 3 Computing likelihoods 4 Illustration and outlook Matthias Birkner (LMU Munich) Banff, Sept / 42

3 Introduction Wright-Fisher model and Kingman s coalescent Wright-Fisher model: The fundamental model for genetic drift A (haploid) population of N individuals per generation, each individual in the present generation picks a parent at random from the previous generation, genetic types are inherited (possibly with a small probability of mutation). past Matthias Birkner (LMU Munich) Banff, Sept / 42

4 Introduction Wright-Fisher model and Kingman s coalescent Wright-Fisher model: The fundamental model for genetic drift A (haploid) population of N individuals per generation, each individual in the present generation picks a parent at random from the previous generation, genetic types are inherited (possibly with a small probability of mutation). past Matthias Birkner (LMU Munich) Banff, Sept / 42

5 Introduction Wright-Fisher model and Kingman s coalescent Wright-Fisher model: The fundamental model for genetic drift A (haploid) population of N individuals per generation, each individual in the present generation picks a parent at random from the previous generation, genetic types are inherited (possibly with a small probability of mutation). past Matthias Birkner (LMU Munich) Banff, Sept / 42

6 Introduction Wright-Fisher model and Kingman s coalescent Wright-Fisher model: The fundamental model for genetic drift A (haploid) population of N individuals per generation, each individual in the present generation picks a parent at random from the previous generation, genetic types are inherited (possibly with a small probability of mutation). past Matthias Birkner (LMU Munich) Banff, Sept / 42

7 Introduction Genealogical point of view Wright-Fisher model and Kingman s coalescent Sample n ( N) individuals from the present generation present past Matthias Birkner (LMU Munich) Banff, Sept / 42

8 Kingman s coalescent Introduction Wright-Fisher model and Kingman s coalescent Theorem (Kingman (& Hudson, Griffiths), 1982) In the limit N, the genealogy of an n- sample, measured in units of N generations, is described by a continuous-time Markov chain where each pair of lineages merges at rate 1. Matthias Birkner (LMU Munich) Banff, Sept / 42

9 Kingman s coalescent Introduction Wright-Fisher model and Kingman s coalescent Theorem (Kingman (& Hudson, Griffiths), 1982) In the limit N, the genealogy of an n- sample, measured in units of N generations, is described by a continuous-time Markov chain where each pair of lineages merges at rate 1. Robustness. The same limit appears for any exchangeable offspring vectors (ν 1,...,ν N ), (independent over generations), if time is measured in units of N σ 2 generations, where σ2 = lim N Var(ν 1) (under a third moment condition on ν 1 ). Matthias Birkner (LMU Munich) Banff, Sept / 42

10 Introduction Wright-Fisher model and Kingman s coalescent Modeling neutral variation: Superimposing types on the coalescent Assume that the considered genetic types do not affect their bearer s reproductive succes. If as population size N, N mutation prob. per ind. per generation r, σ2 the type configuration in the sample can be described by putting mutations with rate r along the genealogy. Matthias Birkner (LMU Munich) Banff, Sept / 42

11 Question Introduction Multiple merger coalescents What if the variability of surviving offspring numbers across individuals is so large that reasonably individual offspring variance σ 2? This might happen e.g. in marine species (so-called reproduction sweepstakes). Matthias Birkner (LMU Munich) Banff, Sept / 42

12 Introduction Multiple merger coalescents Coalescents with multiple collisions, aka Λ-coalescents While n lineages, any k coalesce at rate λ n,k = [0,1] x k 2 (1 x) n k Λ(dx), where Λ is a finite measure on [0,1]. (Sagitov, 1999; Pitman, 1999). Matthias Birkner (LMU Munich) Banff, Sept / 42

13 Introduction Multiple merger coalescents Coalescents with multiple collisions, aka Λ-coalescents While n lineages, any k coalesce at rate λ n,k = [0,1] x k 2 (1 x) n k Λ(dx), where Λ is a finite measure on [0,1]. (Sagitov, 1999; Pitman, 1999). Interpretation: re-write λ n,k = [0,1] xk (1 x) n k 1 x 2 Λ(dx) to see: at rate 1 x 2 Λ([x,x + dx]), an x-resampling event occurs. Thinking forwards in time, this corresponds to an event in which the fraction x of the total population is replaced by the offspring of a single individual. Matthias Birkner (LMU Munich) Banff, Sept / 42

14 Introduction Multiple merger coalescents Coalescents with multiple collisions, aka Λ-coalescents While n lineages, any k coalesce at rate λ n,k = [0,1] x k 2 (1 x) n k Λ(dx), where Λ is a finite measure on [0,1]. (Sagitov, 1999; Pitman, 1999). Interpretation: re-write λ n,k = [0,1] xk (1 x) n k 1 x 2 Λ(dx) to see: at rate 1 x 2 Λ([x,x + dx]), an x-resampling event occurs. Thinking forwards in time, this corresponds to an event in which the fraction x of the total population is replaced by the offspring of a single individual. Form of rates stems from λ n,k = λ n+1,k + λ n+1,k+1 (consistency condition). Matthias Birkner (LMU Munich) Banff, Sept / 42

15 Introduction Multiple merger coalescents Coalescents with multiple collisions, aka Λ-coalescents While n lineages, any k coalesce at rate λ n,k = [0,1] x k 2 (1 x) n k Λ(dx), where Λ is a finite measure on [0,1]. (Sagitov, 1999; Pitman, 1999). Interpretation: re-write λ n,k = [0,1] xk (1 x) n k 1 x 2 Λ(dx) to see: at rate 1 x 2 Λ([x,x + dx]), an x-resampling event occurs. Thinking forwards in time, this corresponds to an event in which the fraction x of the total population is replaced by the offspring of a single individual. Form of rates stems from λ n,k = λ n+1,k + λ n+1,k+1 (consistency condition). Note: Λ = δ 0 corresponds to Kingman s coalescent. Matthias Birkner (LMU Munich) Banff, Sept / 42

16 Introduction Multiple merger coalescents Cannings models in the domain of attraction of a Λ-coalescent Fixed population size N, exchangeable offspring numbers in one generation ( ν1,ν 2,...,ν N ). Sagitov (1999), Möhle & Sagitov (2001) clarify under which conditions the genealogies of a sequence of exchangeable finite population models are described by a Λ-coalescent: c N := pair coalescence probability over one generation 0 ( c N = 1 N 1 E[ν 1(ν 1 1)] ) two double mergers asymptotically negligible compared to one triple merger Nc N Pr ( a given family has size Nx ) 1 x y 2 Λ(dy) Time is measured in 1/c N generations (in general 1/pop. size) Matthias Birkner (LMU Munich) Banff, Sept / 42

17 Introduction The Beta(2 α, α)-class Note: There are many Λ-coalescents. Maybe a natural first candidate (considered by Eldon & Wakeley, Genetics 2006): Λ = wδ 0 + (1 w)δ ψ with w,ψ (0,1) Matthias Birkner (LMU Munich) Banff, Sept / 42

18 Introduction The Beta(2 α, α)-class Note: There are many Λ-coalescents. Maybe a natural first candidate (considered by Eldon & Wakeley, Genetics 2006): Λ = wδ 0 + (1 w)δ ψ with w,ψ (0,1) Simple, few parameters But: No realistic microscopic population model Matthias Birkner (LMU Munich) Banff, Sept / 42

19 Introduction The Beta(2 α, α)-class A heavy-tailed Cannings model and Beta-coalescents Haploid population of size N. Individual i has X i potential offspring, X 1,X 2,...,X N are i.i.d. with mean m := E [ X 1 ] > 1, Pr ( X 1 k ) Const. k α with α (1,2). Note: infinite variance. Sample N without replacement from all potential offspring to form the next generation. Matthias Birkner (LMU Munich) Banff, Sept / 42

20 Introduction The Beta(2 α, α)-class A heavy-tailed Cannings model and Beta-coalescents Haploid population of size N. Individual i has X i potential offspring, X 1,X 2,...,X N are i.i.d. with mean m := E [ X 1 ] > 1, Pr ( X 1 k ) Const. k α with α (1,2). Note: infinite variance. Sample N without replacement from all potential offspring to form the next generation. Theorem (Schweinsberg, 2003) Let c N = prob. of pair coalescence one generation back in N-th model. c N const. N 1 α, measured in units of 1/c N generations, the genealogy of a sample from the N-th model is approximately described by a ( Λ-coalescent with Λ = Beta(2 α, α). ) 1 Beta(2 α, α)(dx) = ½ [0,1] (x) Γ(2 α)γ(α) x1 α (1 x) α 1 dx Matthias Birkner (LMU Munich) Banff, Sept / 42

21 Introduction The Beta(2 α, α)-class The family Beta(2 α, α), α (1, 2] Kingman s coalescent included as boundary case: Beta(2 α,α) δ 0 weakly as α 2. Smaller α means tendency towards more extreme resampling events. For α 1, corresponding coalescents do not come down from infinity Beta(2 α, α)-coalescents appear as genealogies of α-stable continuous mass branching process (via a time-change). Scaling relation of mutation rate per generation relative to population size depends on α! Matthias Birkner (LMU Munich) Banff, Sept / 42

22 Neutral mutations Infinitely many sites Infinitely many sites model (Kimura 1971) Model genetic locus as infinite sequence of completely linked sites, mutations always hit a new site. All mutations are neutral. Mathematical abstraction: a gene is [0,1] a type is a configuration of points on [0,1] Ethier & Griffiths (1987) parametrisation: type space E = [0,1] N mutation operator Bf ( (x 1,x 2,... ) ) = r 1 0 f ( (u,x 1,x 2,... ) ) f ( (x 1,x 2,... ) ) du Matthias Birkner (LMU Munich) Banff, Sept / 42

23 Neutral mutations Infinitely many sites Berestycki, Berestycki & Schweinsberg (2007) and Berestycki, Berestycki & Limic (2009) describe precisely the asymptotic behaviour of the frequency spectrum under a Λ-coalescent Matthias Birkner (LMU Munich) Banff, Sept / 42 Infinitely many sites model (Kimura 1971) Model genetic locus as infinite sequence of completely linked sites, mutations always hit a new site. All mutations are neutral. Mathematical abstraction: a gene is [0,1] a type is a configuration of points on [0,1] Ethier & Griffiths (1987) parametrisation: type space E = [0,1] N mutation operator Bf ( (x 1,x 2,... ) ) = r 1 0 f ( (u,x 1,x 2,... ) ) f ( (x 1,x 2,... ) ) du

24 Neutral mutations Infinitely many sites Combinatorics of the Infinitely-many-sites model Model genetic locus as infinite sequence of completely linked sites, mutations always hit a new site Example: segr. site Seq (0=wild type, 1=mutant assume known ancestral types) Obs. fit IMS no sub-matrix (and no row permutation) a b c a b c Matthias Birkner (LMU Munich) Banff, Sept / 42

25 Neutral mutations Infinitely many sites Combinatorics of the infinitely-many-sites model, II If the infinitely-many-sites model applies, the observations correspond to a unique rooted perfect phylogeny (or genetree ). Sequences, Genetree, obs. types segr. site Seq ,4 5 Construct e.g. using Gusfield s (1991) algorithm. 4 3 type multiplicity (1, 0) 1 (2, 1, 0) 1 (4, 3, 0) 2 (3, 0) 1 Note: purely combinatorial, does not depend on a probabilistic model for the observations. Matthias Birkner (LMU Munich) Banff, Sept / 42

26 Neutral mutations (Λ-) Ethier-Griffiths urn Simulating samples under the IMS model Literally, the data arises (in our model) in a stochastic two-step procedure First generate genealogy acc. to coalescent. Then superimpose mutations. Matthias Birkner (LMU Munich) Banff, Sept / 42

27 Neutral mutations (Λ-) Ethier-Griffiths urn Simulating samples under the IMS model Literally, the data arises (in our model) in a stochastic two-step procedure First generate genealogy acc. to coalescent. Then superimpose mutations. The Ethier-Griffiths urn (1987) can be used to generate a random sample of size n under Kingman s coalescent (with mutation rate r per line) in one pass : Start with 2 leaves. When there are k leaves: Add a mutation to a leaf w. prob. 2r 2r+(k 1), split one leaf w. prob. k 1 2r+(k 1) (leaf picked uniformly among the k). Stop when n + 1 leaves, delete last leaf. Matthias Birkner (LMU Munich) Banff, Sept / 42

28 Neutral mutations (Λ-) Ethier-Griffiths urn Simulating samples under the IMS model: Λ-case (Y (n) t ) 0 block counting process of Λ-coalescent starting from n blocks: Jump from i to j {1,2,...,i 1} at rate q ij := ( i i j+1) λi,i j+1. τ 1 := inf{t 0 : Y (n) t = 1}. Matthias Birkner (LMU Munich) Banff, Sept / 42

29 Neutral mutations (Λ-) Ethier-Griffiths urn Simulating samples under the IMS model: Λ-case (Y (n) t ) 0 block counting process of Λ-coalescent starting from n blocks: Jump from i to j {1,2,...,i 1} at rate q ij := ( i i j+1) λi,i j+1. Ỹ (n) t τ 1 := inf{t 0 : Y (n) t = 1}. := Y (n) (τ 1 t) time-reversed block counting process (Ỹ (n) t = for t τ 1 ). Jump rates q (n) ji = g niq i j g nj, q (n) n = q nn = n 1 j=1 q nj (Nagasawa s formula) P(Ỹ (n) 0 = k) = P(Y (n) τ 1 = k) = g nkq k1. g ni := E t = i)dt is the Green function (in general, not known explicitly, but easy recursion). 0 1(Y (n) Matthias Birkner (LMU Munich) Banff, Sept / 42

30 Neutral mutations (Λ-) Ethier-Griffiths urn Simulating samples under the IMS model: Λ-case, cont. The n-λ- Ethier-Griffiths urn (mutation rate r). Begin with K leaves, P(K = k) = P(Ỹ (n) 0 = k). While there are k leaves: Add a mutation to a leaf w. prob. split one leaf into l if k = n goto stop (leaf picked uniformly among the k). Stop. w. prob. eq(n) w. prob. r kr eq (n) kk, k,k+l 1 eq (n) kk eq nn (n) kr eq nn (n), Matthias Birkner (LMU Munich) Banff, Sept / 42

31 Computing likelihoods Tree probabilities Recursion for tree probabilities Can calculate P (r,λ) ( observed sequence data (t,n) ) =: p(t,n) via Matthias Birkner (LMU Munich) Banff, Sept / 42

32 Computing likelihoods Recursion for tree probabilities Tree probabilities Can calculate P (r,λ) ( observed sequence data (t,n) ) =: p(t,n) via p(t,n) = 1 rn + λ n r + rn + λ n + n i i:n i 2 k=2 r 1 rn + λ n d ( n k i:n i =1,x i0 unique, s(x i ) x j j i:n i =1, x i0 unique ) λ n,k n i k + 1 n k + 1 p( t,n (k 1)e i ) p ( s i (t),n ) ( (n j + 1)p ri (t), r i (n + e j ) ). j:s(x i)=x j Extends Ethier & Griffiths (1987) to Λ-coalescents and Möhle s recursion (2005) to IMS model (and is a true recursion in the complexity of the sample). Matthias Birkner (LMU Munich) Banff, Sept / 42

33 Computing likelihoods Tree probabilities Compute probabilities of observed data naïve approach: simulate (often), est. via empirical frequency of observed data (infeasible: req. prob. may be ) Use exact recursions for moderate sample complexities. Approach more complex samples by version of Griffiths & Tavaré s (1994) Monte Carlo method [ τ 1 p(t,n) = E (t,n) i=0 ] f (r,λ) (X i ) For suitable Markov chain X i on sample configurations. Estimate expectation via empirical mean of independent runs. extension to Λ-coalescents by B. & Blath (2008) Matthias Birkner (LMU Munich) Banff, Sept / 42

34 Computing likelihoods Tree probabilities Analyzing a simulated sample of size 15: sd b estimate ) log10( Griffiths & Tavaré log10(#runs) Matthias Birkner (LMU Munich) Banff, Sept / 42

35 Histories Computing likelihoods Importance sampling Interpret genealogy as sequence of historical states: H = ( H τ = ((1),(0)),H τ 1,...,H 1,H 0 = (t,n) ) ( ((3,1,0),(1,0),(2,1,0),(0) ),(1,2,1,1) ) Matthias Birkner (LMU Munich) Banff, Sept / 42

36 Histories Computing likelihoods Importance sampling Interpret genealogy as sequence of historical states: H = ( H τ = ((1),(0)),H τ 1,...,H 1,H 0 = (t,n) ) 0 H 6 1 H 5 H H 3 H 2 H 1 H 0 ( ((3,1,0),(1,0),(2,1,0),(0) ),(1,2,1,1) ) Matthias Birkner (LMU Munich) Banff, Sept / 42

37 Histories Computing likelihoods Importance sampling Interpret genealogy as sequence of historical states: H = ( H τ = ((1),(0)),H τ 1,...,H 1,H 0 = (t,n) ) ( ((3,1,0),(1,0),(2,1,0),(0) ),(1,2,1,1) ) Matthias Birkner (LMU Munich) Banff, Sept / 42

38 Histories Computing likelihoods Importance sampling Interpret genealogy as sequence of historical states: H = ( H τ = ((1),(0)),H τ 1,...,H 1,H 0 = (t,n) ) ( ((3,1,0),(1,0),(2,1,0),(0) ),(1,2,1,1) ) Matthias Birkner (LMU Munich) Banff, Sept / 42

39 Histories Computing likelihoods Importance sampling Interpret genealogy as sequence of historical states: H = ( H τ = ((1),(0)),H τ 1,...,H 1,H 0 = (t,n) ) different histories can lead to same sample ( ((3,1,0),(1,0),(2,1,0),(0) ),(1,2,1,1) ) Matthias Birkner (LMU Munich) Banff, Sept / 42

40 Importance sampling Computing likelihoods Importance sampling We have p(t,n) = P (r,λ) ( H0 = (t,n) ) = = H:H 0 =(t,n) H:H 0 =(t,n) ( ) P (r,λ) H Q ( H ) Q(H), }{{} =:w(h) importance weight P (r,λ) ( H ) for any law Q on histories s.th. P (r,λ) {H0 =(t,n)} Q. Thus, p(t,n) 1 R R w(h (i) ), i=1 where H (1),..., H (R) are independent samples from Q Matthias Birkner (LMU Munich) Banff, Sept / 42

41 Computing likelihoods Importance sampling (Theoretical) optimal solution p(t,n) = H:H 0 =(t,n) ( ) P (r,λ) H Q ( H ) Q( H ) 1 R R w(h (i) ) i=1 Q opt ( ) := P (r,λ) ( H 0 = (t,n) ) is optimal: Variance of estimator is zero since w(h (i) ) p(t,n). Finding Q opt is as hard as the original problem. H 0,H 1,... is Markov chain under Q opt (time reversal of Λ-Ethier-Griffiths urn). Remark: Transition probabilities q GT (H i H i+1 ) P (r,λ) (H i+1 H i ) gives (Λ-)Griffiths-Tavaré method. Matthias Birkner (LMU Munich) Banff, Sept / 42

42 Computing likelihoods Importance sampling Stephens and Donnelly s (2000) IMS candidate Kingman case: Choose individual uniformly: If type is unique in sample, remove outmost mutation, if at least two individuals with this type, merge two lines. (this would be optimal for parent-independent mutations) Heuristic extension to Λ case: ( si (t),n ) w.p. 1 if n i = 1, x i0 unique, s i (x i ) x j j ( (t,n) ri (t,r i (n + e j )) w.p. 1 if n i = 1, x i0 unique, s i (x i ) x j ( ) t,n (k 1)ei where q ni (k) = w.p. n i q ni (k) if 2 k n i, q n,n k+1 P ni, jump probabilities of block counting process. l=2 q n,n l+1 Matthias Birkner (LMU Munich) Banff, Sept / 42

43 Computing likelihoods Importance sampling Analyzing a simulated sample of size 15: sd b estimate ) log10( Griffiths & Tavaré Stephens & Donnelly log10(#runs) Matthias Birkner (LMU Munich) Banff, Sept / 42

44 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea Sample of size n where exactly one mutation is visible (in d copies). 0 n-d 1 d { } p (1) (r,λ) (n,d) = P most recent event involves individual bearing (r,λ) mutation Probability can be computed Kingman case: explicit formula (HUW (2008)) Λ case: numerically, using recursion Matthias Birkner (LMU Munich) Banff, Sept / 42

45 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. For a general sample (t,n) put i u i,m = { where mutation m is present in d m individuals. Matthias Birkner (LMU Munich) Banff, Sept / 42

46 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. For a general sample (t,n) carrying: = d put i u i,m = { p (1) (r,λ) (n,d m) ni d m if i bears m where mutation m is present in d m individuals. Matthias Birkner (LMU Munich) Banff, Sept / 42

47 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. For a general sample (t,n) carrying: put i u i,m = { p (1) (r,λ) (n,d m) ni d m if i bears m where mutation m is present in d m individuals. Matthias Birkner (LMU Munich) Banff, Sept / 42

48 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. For a general sample (t,n) carrying: not carrying: put i { (1) p (r,λ) u i,m = (n,d m) ni d ( m (1) 1 p (r,λ) (n,d m) ) n i n d m where mutation m is present in d m individuals. if i bears m otherwise, Matthias Birkner (LMU Munich) Banff, Sept / 42

49 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. For a general sample (t,n) carrying: not carrying: put i { (1) p (r,λ) u i,m = (n,d m) ni d ( m (1) 1 p (r,λ) (n,d m) ) n i n d m where mutation m is present in d m individuals if i bears m otherwise, Matthias Birkner (LMU Munich) Banff, Sept / 42

50 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. For a general sample (t,n) carrying: not carrying: put i { (1) p (r,λ) u i,m = (n,d m) ni d ( m (1) 1 p (r,λ) (n,d m) ) n i n d m if i bears m otherwise, where mutation m is present in d m individuals. Propose type i according to { ( ) q Λ-HUW i (t,n) m u i,m if i is allowed to act 0 otherwise. Matthias Birkner (LMU Munich) Banff, Sept / 42

51 Computing likelihoods Importance sampling Hobolth, Uyenoyama & Wiuf s (2008) idea contd. If proposed type i is singleton: remove outmost mutation, has n i 2: merger inside type i. Kingman case: merge two lines Λ-case: propose l + 1-merger w.p. P r,λ { 0 n 1 d o l 0 n 1 d o } Matthias Birkner (LMU Munich) Banff, Sept / 42

52 Computing likelihoods Importance sampling Analyzing a simulated sample of size 15: sd b estimate ) log10( Griffiths & Tavaré Stephens & Donnelly Hobolth, Uyenoyama & Wiuf log10(#runs) Matthias Birkner (LMU Munich) Banff, Sept / 42

53 Computing likelihoods Using Pairs of Mutations Importance sampling For sample (t,n) denote possible steps by { {decrease n i by l} if n i > 1,l > 0, (i,l) := {remove o(i)} if n i = 1,l = 0 and o(i) leaf. 0 0 m(i) m(i) 1 6 i Let m(i) be type corresponding to i in induced subtree. Set { } decrease multiplicity of m(i) by n P (r,λ) i l in {m v {m1,m 2 }(i,l) = 1, m 2 }-genetree #m(i) if l > 1, { } last event origin of o(m(i)) in P (r,λ) l = 0. {m 1, m 2 }-genetree 4 2 Matthias Birkner (LMU Munich) Banff, Sept / 42

54 Computing likelihoods Using Pairs of Mutations cont d Importance sampling Summing over all pairs of mutations turned out to be inefficient. Sum over pairs where one mutation is o(i). For general (t,n) choose step (i,l) according to { m o(i) Q Λ-HUW2 (i,l t,n) v {o(i),m}(i,l) if i is allowed to act, 0 otherwise. Matthias Birkner (LMU Munich) Banff, Sept / 42

55 Computing likelihoods Importance sampling Analyzing a simulated sample of size 15: sd b estimate ) log10( Griffiths & Tavaré Stephens & Donnelly Hobolth, Uyenoyama & Wiuf Hobolth, Uyenoyama & Wiuf II log10(#runs) Matthias Birkner (LMU Munich) Banff, Sept / 42

56 Performance Computing likelihoods Importance sampling Simulated 100 samples of size 15 with random r and α. Analysed with (new) random r and α. Time needed to get relative error below 0.01: (a) measured in log10(# runs of MC) (b) measured in log10(seconds) black = Λ-GT, green = Λ-SD, red = Λ-HUW, blue = Λ-HUW-II Matthias Birkner (LMU Munich) Banff, Sept / 42

57 Computing likelihoods Importance sampling Remarks Potential for speed-up: Compute likelihoods for several parameter sets (r 1,Λ 1 ),...,(r m,λ m ) simulataneously using sampled histories H (1),..., H (R) from one driving value (r 0,Λ 0 ) Pre-compute and store p (r,λ) (t,n)) for small trees (e.g. at most m mutations), stop chain when entering set of small trees Matthias Birkner (LMU Munich) Banff, Sept / 42

58 Computing likelihoods Importance sampling A simulation study: Comparing three estimators (sample size n = 100) Three (ML) estimators for α α SS based on summary statistic (no. mutations,no. singletons) (computation fast via recursion, Eldon & Wakeley s approach) α data based on full data α fulltree based on knowledge of full genealogical tree (in reality: unknown) Matthias Birkner (LMU Munich) Banff, Sept / 42

59 Computing likelihoods Importance sampling A simulation study: Comparing three estimators (sample size n = 100) Three (ML) estimators for α α SS based on summary statistic (no. mutations,no. singletons) (computation fast via recursion, Eldon & Wakeley s approach) α data based on full data α fulltree based on knowledge of full genealogical tree (in reality: unknown) True α = 1 α SS α data α fulltree mean MSE True α = 1.4 α SS α data α fulltree mean MSE True α = 2 α SS α data α fulltree mean MSE (Estimated values, based on 100 replicates. True r = 1.0) Matthias Birkner (LMU Munich) Banff, Sept / 42

60 Computing likelihoods Importance sampling Estimating aspects of the genealogy Method allows to estimate times, conditional on data, e.g. E (r,λ) [ TMRCA of sample obs. data = (t,n) ], P (r,λ) ( TMRCA of sample t obs. data = (t,n) ) : For any test function ϕ E (r,λ) [ ϕ(h) obs. data = (t,n) ] 1 R R i=1 ϕ(h (i) ) P ( (r,λ) H (i) ) Q ( H (i)) with H (1),...,H (R) i.i.d. draws from Q( ) (compatible with P (r,λ) ( obs. data)). (Literally: Plus additional independent exponential waiting times between events in the genealogy) Matthias Birkner (LMU Munich) Banff, Sept / 42

61 Illustration and outlook Illustration Dataset from Ward et al, Extensive Mitochondrial Diversity Within a Single Amerindian Tribe, PNAS 1991 Analysis with Beta-Coalescent: r e 25 1e 24 1e 23 1e e 21 1e 20 2e 204e 20 9e 20 8e 20 6e α Mitochondrial control region from 55 female Nuu-Chah-Nulth: α ML = 1.9, r ML = 2.2 (Sample as edited in Griffiths & Tavaré, Stat. Sci., 1994) 1e 21 1e 22 1e 23 1e 24 1e 25 Matthias Birkner (LMU Munich) Banff, Sept / 42

62 r r Illustration and outlook r r Illustration r r Genetic variation at the mitochondrial cyt b locus of Atlantic cod: log-likelihood surfaces Carr & Marshall 1991 Pepin & Carr 1993 Árnason & Palsson α α α bα ML = 1.4,br ML = 0.8, bα BBS = 1.4 bα ML = 1.4,br ML = 0.6, bα BBS = 1.5 bα ML = 1.65,br ML = 0.7, bα BBS = Árnason et al 1998 Árnason et al 2000 Sigurgíslason & Árnason α α α bα ML = 1.55,br ML = 0.6, bα BBS = 1.8 bα ML = 1.4,br ML = 0.8, bα BBS = 1.5 bα ML = 1.4,br ML = 0.8, bα BBS = Matthias Birkner (LMU Munich) Banff, Sept / 42

63 Illustration and outlook Illustration α-effective population size can the figures make sense? In Schweinsberg s model, we have pair coalescence prob. c N C N 1 α (C = αγ(α)γ(2 α)m α, where m = mean of X 1 ) Matthias Birkner (LMU Munich) Banff, Sept / 42

64 Illustration and outlook Illustration α-effective population size can the figures make sense? In Schweinsberg s model, we have pair coalescence prob. c N C N 1 α (C = αγ(α)γ(2 α)m α, where m = mean of X 1 ) Let µ = mutation rate per gen. at the considered locus, then µ c N r Matthias Birkner (LMU Munich) Banff, Sept / 42

65 Illustration and outlook Illustration α-effective population size can the figures make sense? In Schweinsberg s model, we have pair coalescence prob. c N C N 1 α (C = αγ(α)γ(2 α)m α, where m = mean of X 1 ) Let µ = mutation rate per gen. at the considered locus, then µ r, ( c N hence rαγ(α)γ(2 α)m α) 1/(α 1). N eff,α µ Matthias Birkner (LMU Munich) Banff, Sept / 42

66 Illustration and outlook Illustration α-effective population size can the figures make sense? In Schweinsberg s model, we have pair coalescence prob. c N C N 1 α (C = αγ(α)γ(2 α)m α, where m = mean of X 1 ) Let µ = mutation rate per gen. at the considered locus, then µ r, ( c N hence rαγ(α)γ(2 α)m α) 1/(α 1). N eff,α µ Using µ = (Árnason, 2004), α = 1.5, r = 1 (and, ad hoc, m = 2), this gives N eff,α= Árnason (2004) writes:... the actual population size [of atlantic cod] is not < 10 9 and probably one or two orders of magnitude larger. Matthias Birkner (LMU Munich) Banff, Sept / 42

67 Illustration and outlook Illustration α-effective population size can the figures make sense? In Schweinsberg s model, we have pair coalescence prob. c N C N 1 α (C = αγ(α)γ(2 α)m α, where m = mean of X 1 ) Let µ = mutation rate per gen. at the considered locus, then µ r, ( c N hence rαγ(α)γ(2 α)m α) 1/(α 1). N eff,α µ Using µ = (Árnason, 2004), α = 1.5, r = 1 (and, ad hoc, m = 2), this gives N eff,α= Árnason (2004) writes:... the actual population size [of atlantic cod] is not < 10 9 and probably one or two orders of magnitude larger. By contrast, using a Wright-Fisher model and N eff,α=2 µ = r, we have (using r α=2 = 2): Neff,α= Matthias Birkner (LMU Munich) Banff, Sept / 42

68 Summary & Outlook Illustration and outlook Outlook Eldon & Wakeley, Genetics 2006, wrote For many species, the coalescent with multiple mergers might be a better null model than Kingman s coalescent. Matthias Birkner (LMU Munich) Banff, Sept / 42

69 Illustration and outlook Outlook Summary & Outlook Eldon & Wakeley, Genetics 2006, wrote For many species, the coalescent with multiple mergers might be a better null model than Kingman s coalescent. For panmictic fixed-size discrete generations populations, haploid neutral one-locus theory is mathematically complete. Tools for estimation exist, results point towards non-kingman-ness in certain cases. Statistical properties of estimators? (Further) speed-up of computer-intensive methods? A good class of alternative models? In particular, true diploid models? Application to scenarios with selection? Matthias Birkner (LMU Munich) Banff, Sept / 42

70 Illustration and outlook Outlook Thank you for your attention! Matthias Birkner (LMU Munich) Banff, Sept / 42