Intro Likelihood & coa MCMC IS Sim tests Conclusions

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1 A faire RL : homogénéiser les notations avec les macros de FR, remplacer IS par SIS partout, insister sur le sequential = le long de la construction de l arbre, mieux séparer l obtention de la récurrence, les pi exactes et les pi chapeaux, mettre en bleu les mots clés ajouter l terme p(htau)=distribution stationnaire des états alléliques partout ou c est nécessaire FR : 1) Faire backward dans un cours précédent... (et d autres trucs sur ma présentation de backward) 2) l opérateur différentiel φ j n est pas explicité = phrases plus compliquées

2 Module de Master 2 Biostatistique: modèles de génétique des populations Likelihood-based demographic inference using the coalescent Raphaël Leblois & François Rousset Centre de Biologie pour la Gestion des populations (CBGP, Montpellier) Institut des Sciences de l Evolution, (ISEM, Montpellier) Janvier 2017

3 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Metropolis-Hastings MsVar example Conclusions on MCMC Griffiths et al. s IS Griffiths et al s recursion Old IS scheme New IS scheme A general method based on diffusion Approximations of π Simulation tests Precision Validation Robustness MCMC vs. IS Conclusions

4 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Griffiths et al. s IS Simulation tests Conclusions

5 Intro Likelihood & coa MCMC IS Sim tests Conclusions Typical biological question : There are demographic evidences that orang-utan population sizes have collapsed but what is the major cause of the decline, when did it start and how strong is it? Can population genetics help? - Can we infer the time of the event? - Can we infer the strength of the population size decrease?

6 Methods based on coalescence simulations (Reminder...) Genealogy of the population Genealogy of the sample Coalescent tree forward in time backward in time ; ; P(T k = t) k(k 1) k(k 1) 2N e t 2N P(m t) = (µt)m e µt m!

7 Two different ways to use the coalescent theory Exploratory approaches & simulation tests - The coalescent allows efficient simulations of the genetic variability under various demo-genetic models (sample vs. population) Specify the model and parameter values Coalescent process Simulated data sets Inferential approach - The coalescent allows the inference of populationnal evolutionary parameters (genetic, demographic, reproductive,...), some of those methods uses all the information contained in the genetic data (likelihood-based methods) a real data set Coalescent process infer the model parameters

8 Two different ways to use the coalescent theory Exploratory approaches & simulation tests - The coalescent allows efficient simulations of the genetic variability under various demo-genetic models (sample vs. population) Specify the model and parameter values Coalescent process Simulated data sets Inferential approach - The coalescent allows the inference of populationnal evolutionary parameters (genetic, demographic, reproductive,...), some of those methods uses all the information contained in the genetic data (likelihood-based methods) a real data set Coalescent process infer the model parameters

9 Likelihood-based inference under the coalescent Inferential approaches are based on the modeling of population genetic processes. Each population genetic model is characterized by a set of demographic and genetic parameters P The aim is to infer those parameters from a polymorphism data set (i.e. a genetic sample) The genetic sample is then considered as the realization ( output ) of a stochastic process defined by the demo-genetic model

10 Likelihood-based inference under the coalescent First, compute or estimate the likelihood L(P ; D), i.e. the probability P(D; P ) of observing the data D for some parameter values P Second, infer the likelihood surface over all parameter values, find the set of parameter values that maximize it, and compute CI (maximum likelihood method), or Compute posterior distributions and compare with priors (Bayesian approach).

11 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Griffiths et al. s IS Simulation tests Conclusions

12 Likelihood computations under the coalescent Problem : Most of the time, the likelihood P(D; P ) of a genetic sample cannot be computed because there is no explicit mathematical expression However, the probability P(D; P G k ) of observing the data D given a specific genealogy G k can be computed for some parameter values P. Then we take the sum of all genealogy-specific likelihoods on the whole genealogical space, weighted by the probability of the genealogy given the parameters : L(P ; D) = G P(D; P G)P(G; P ) dg

13 Likelihood computations under the coalescent The likelihood can be written as the sum of P(D; P G k ) over the genealogical space (all possible genealogies) : L(P Mutation ; D) = P(D; P G)P(G; P ) dg G Demography (Coalescent) Genealogies are missing data, they are important for the computation of the likelihood but there is no interest in estimating them. very different from the phylogenetic approaches

14 Likelihood computations under the coalescent The likelihood can be written as the sum of P(D; P G k ) over the genealogical space (all possible genealogies) : L(P ; D) = G P(D; P G)P(G; P ) dg...usually impossible to sum over all possible genealogies... Monte Carlo simulations are used : a large number K of genealogies are simulated according to P(G; P ) and the mean over those simulations is taken as the expectation of P(D; P G) : L(P ; D) = E P(G;P )(P(D; P G)) 1 K K P(D; P G k ) k=1

15 Likelihood computations under the coalescent The likelihood can be written as the sum of P(D; P G k ) over the genealogical space (all possible genealogies) : L(P ; D) = G P(D; P G)P(G; P ) dg...usually impossible to sum over all possible genealogies... Monte Carlo simulations are used : L(P ; D) = E P(G;P )(P(D; P G)) 1 K K P(D; P G k ) k=1 many many genealogies necessary for a good estimation of the likelihood...

16 Likelihood computations under the coalescent Monte Carlo simulations are used : L(P ; D) = E P(G;P )(P(D; P G)) 1 K K P(D; P G k ) k=1 Monte Carlo simulations are often not very efficient because there are too many genealogies giving extremely low probabilities of observing the data, more efficient algorithms are used to explore the genealogical space and focus on genealogies well supported by the data.

17 Likelihood computations under the coalescent Two main approaches developed using more efficient algorithms that allows better exploration of the genealogies proportionally to their probability of explaining the data P(D; P G). MCMC Monte Carlo Markov chains on the genealogical and the parameter space, based on Felsenstein s pruning algorithm (1973,1981) Felsenstein, J. (1981). Evolutionary trees from DNA sequences : A maximum likelihood approach. J. of Mol. Evol. 17 (6) : IS Importance Sampling on genealogies, based on the work of Griffiths & Tavaré Griffiths, R.C. and S. Tavaré (1994). Simulating probability distributions in the coalescent. Theor. Pop. Biol., 46 :

18 Likelihood computations under the coalescent More efficient algorithms that allows better exploration of the genealogies proportionally to their probability of explaining the data P(D; P G) MCMC Felsenstein s pruning algorithm. - Easier to implement, can easily consider various models - Implemented in many softwares (LAMARC, Batwing, MsVar, MIGRATE, IM) IS Griffiths &Tavaré s coalescent recursion - Extension to different models may be difficult - Implemented in fewer softwares (Genetree, Migraine)

19 Likelihood computations under the coalescent More efficient algorithms that allows better exploration of the genealogies proportionally to their probability of explaining the data P(D; P G) MCMC Felsenstein s pruning algorithm (quick overview) - Easier to implement, can consider various models - Implemented in many softwares (LAMARC, Batwing, MsVar, MIGRATE, IM) IS Griffiths &Tavaré s coalescent recursion - Extension to different models may be difficult - Implemented in fewer softwares (Genetree, Migraine)

20 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Griffiths et al. s IS Simulation tests Conclusions

21 The approach of Felsenstein et al. Based on (1) on the availability of approximate exponential distributions for time intervals between events (coalescence and migration and recombinaison) and (2) on the separation of demographic and mutational processes : 1 The probability of a genealogy given the parameters of the demographic model P(G k ; Pdemo ) can be computed from the distributions of time between events. 2 The probability of the data given a genealogy and mutational parameters P(D; Pmut G k ) can be computed from the mutation model parameters, the mutation rate, tree topology and branch lengths.

22 The approach of Felsenstein et al. Based on (1) on the availability of approximate exponential distributions for time intervals between events (coalescence and migration and recombinaison) and (2) on the separation of demographic and mutational processes : 1 P(G k ; Pdemo ) computed from the distributions of time between events. 2 P(D; Pmut G k ) computed from the mutation parameters, tree topology and branch lengths. From this, an efficient algorithm to explore the genealogical and the parameter spaces should allow the inference of the likelihood over the two spaces. MCMC

23 Felsenstein et al. s MCMC Metropolis-Hastings MsVar example Conclusions on MCMC

24 MCMC with Metropolis-Hastings sampler Full conditional distributions can not be computed, MCMC classical sampler can not thus be used (e.g. Gibbs) Monte Carlo Markov Chains (MCMC) simulations using the Metropolis-Hastings (MH) algorithm - To explore the genealogy space (G) - and the parameter space (P = P demo + P mut ) all algorithms based on the Felsenstein et al. approach uses similar MH/MCMC algorithms with slight differences in the MCMC update steps.

25 Metropolis-Hastings sampling for the coalescent For the Metropolis-Hastings algorithm, we need to compute the ratio of the probability of the proposed update over the current state : 1. Computation of P(G k ; P demo ) : P(G k ; P demo ) = MRCA 1 i=0 γ(t i+1 )e t i+1 t γ(t)dt i - Example for a stable WF population (coalescence only, time homogeneous) MRCA 1 P(G k ; P demo ) = i=0 k i+1 (k i+1 1) e (t i+1 t i ) k i+1 (k i+1 1) 2 2

26 Metropolis-Hastings sampling for the coalescent 1. Computation of P(G k ; P demo ) : P(G k ; P demo ) = MRCA 1 i=0 γ(t i+1 )e t i+1 t γ(t)dt i 2 Then compute the probability P(D; P mut G k ) of the data D given the genealogy G k, by going from the MRCA to the leaves and considering the probability of occurrence of all mutations on each branch of length t b and their effects (i.e.transition among genetic states x y) :

27 Metropolis-Hastings sampling for the coalescent 2 Then compute the probability P(D; P mut G k ) : Mutation matrix : transition probability between genetic states (x, y) P(D; P mut G k ) = = nb branch b=1 effect of mutations P(y x, m b ) Poisson probability for the m b mutations number of mutations P(m b t b ) 2(n 1) ((Mat mut ) m b (µt ) b ) m b e µt b x,y b=1 m b!

28 Metropolis-Hastings sampling for the coalescent 1. Computation of P(G k ; P demo ) : P(G k ; P demo ) = 2 Then compute P(D; P mut G k ) : P(D; P mut G k ) = MRCA 1 i=0 γ(t i+1 )e t i+1 t γ(t)dt i 2(n 1) ((Mat mut ) m b (µt ) b ) m b e µt b x,y b=1 m b! 3 These probabilities are plugged into the MH formula for acceptance probabilities of candidate changes for the next state of the Markov chain. Reminder : P(D; P G k ) = P(D; P mut G k )P(G k ; P demo )

29 Metropolis-Hastings sampling for the coalescent for each update, the new state (P or G ) is accepted or rejected according to the Metropolis-Hastings ratio, the MH ratio is chosen so that the chain converge towards the good stationary distribution P(D; P), e.g. r MH = P(D; P G)Prior(P ) P(P P) P(D; P G)Prior(P) P(P P )

30 Felsenstein et al. s MCMC Metropolis-Hastings MsVar example Conclusions on MCMC

31 Intro Likelihood & coa MCMC IS Sim tests Conclusions Coalescent-based MCMC example : MsVar One example of a coalescent-based MCMC algorithm : MsVar Beaumont, M Detecting Population Expansion and Decline Using Microsatellites. Genetics. Biological contexte : Past changes in population sizes (cf. Orang-Utans) - Details of the demographic and mutation models - few results on the Orang-Utan data set

32 Coalescent-based MCMC example : MsVar Demographic model : a single isolated panmictic (WF) population with a exponential past change in population size.

33 Coalescent-based MCMC example : MsVar Demographic model : a single isolated panmictic (WF) population with a exponential past change in population size. Population contraction or expansion

34 Coalescent-based MCMC example : MsVar Demographic model : a single isolated panmictic (WF) population with a exponential past change in population size. 3 demographic parameters : N, T, N anc + 1 mutation parameter µ 3 scaled parameters (diffusion approx.) : θ, D, θ anc

35 Coalescent-based MCMC example : MsVar Mutation model : Stepwise Mutation Model (SMM)

36 Coalescent-based MCMC example : MsVar P = N, T, N anc, µ P scaled = θ, D, θ anc Aim : infer those parameters (P or P scaled ) from a unique actual genetic sample using coalescent-based MCMC algorithms

37 MH/MCMC of MsVar 1. Initialization step : Build a genealogy that is compatible with the data Starting with the sample, choose a set of events depending on starting values of the parameters ; the events are also chosen to be compatible with the data 2. MCMC steps : Explore the parameter and the genealogical space Update the parameters for population sizes (θ act, D, θ anc ). or Update the genealogy (sequence and times of coalescence and mutation events (T i )) both updates made using the Metropolis-Hastings algorithm

38 MCMC updates in MsVar T i = times of coa & mut, r = θact θ anc pop size ratio, t f = D time of pop size change M. Beaumont : This scheme was devised by trial and error to obtain good rates of convergence.

39 Analyses of MsVar results First check that the chains mixed and converged properly Visual check (very useful) Traces of likelihood / parameters Autocorrelation Compute convergence criteria among chains (GR,...) not always useful... Run different chains and check concordance between results Problem : Convergence is often pretty bad with such coalescent-based MCMC algorithms... but simulation tests show that posterior distributions are generally correct (at least the mode as point estimate) despite no clear convergence indices...

40 Analyses of MsVar results Bayesian method compare posteriors (plain) and priors (dashed)... and test different priors

41 Analyses of MsVar results Bayesian method compute Bayes factor to check for contraction or expansion signal (Posterior prob. model 1) (Prior prob. model 2) BF = (Posterior prob. model 2) (Prior prob. model 1) Equal priors for models 1 and 2, the Bayes factor for a contraction is thus BF = Posterior P(N anc/n act > 1) Posterior P(N anc /N act < 1) BF = # MCMC steps where (N anc/n act > 1) # MCMC steps where (N anc /N act < 1)

42 Intro Likelihood & coa MCMC IS Sim tests An application of MsVar : Orang-Utans and the deforestation of Borneo Does the genome of Orang-utans carry the signature of population bottlenecks? (Goossens et al PLoS Biology) Conclusions

43 An application of MsVar : Orang-Utans and the deforestation of Borneo Population sizes have collapsed : what is the cause? Can population genetics help? (Delgado & Van Schaik, 2001 Evol. Anthropology)

44 An application of MsVar : Orang-Utans and the deforestation of Borneo The data

45 An application of MsVar : Orang-Utans and the deforestation of Borneo MsVar results MsVar efficiently detects a past decrease in population size

46 An application of MsVar : Orang-Utans and the deforestation of Borneo MsVar results FE : beginning of massive forest exploitation F : first farmers HG : first hunter-gatherers MsVar efficiently detects a past decrease in population size and allows for the dating of the beginning of the decrease : massive forest exploitation seems to be the most likely cause

47 Felsenstein et al. s MCMC Metropolis-Hastings MsVar example Conclusions on MCMC

48 Conclusions about MsVar/ MCMC approaches Coalescent theory provides a powerful framework for statistical inference Allows to infer past history from a unique actual sample! (it was impossible with moment based methods) Gene genealogies are missing data (but important...) MCMCs with coalescent simulations are difficult (to run) But what is the robustness to model assumptions : Mutational processes (e.g. large mutation steps long branches) Population structure (e.g. immigrants long branches)

49 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Griffiths et al. s IS Simulation tests Conclusions

50 Likelihood computations under the coalescent More efficient algorithms that allows better exploration of the genealogies (i.e. proportionally to P(D; P G)). MCMC Felsenstein s pruning algorithm. - Easier to implement, can consider various models - Implemented in many softwares (LAMARC, Batwing, MsVar, MIGRATE, IM) IS Griffiths &Tavaré s coalescent recursion (cf. Ewens recursion) - Extension to different models may be difficult - Implemented in fewer softwares (Genetree, Migraine)

51 The approach of Griffiths et al. Coalescent-based likelihood at a given point of the parameter space is an integral aver all possible histories (genealogies with mutations) leading to the present genetic sample Monte Carlo scheme used to compute this integral Histories are build backward in time, event by event, starting from the present sample But computation of exact backward transition probabilities is often too difficult an IS scheme is used to compute the likelihoods by simulation

52 Griffiths et al. s IS Griffiths et al s recursion Old IS scheme New IS scheme A general method based on diffusion Approximations of π

53 Recursions for sampling distributions Ewens 1972 : a Wright-Fisher, infinite-allele model General recursion at stationarity [with e j = (0,, 0, 1 jth, 0,, 0)] : P n (a) = θ n 1 + θ P n 1(a e 1 )+ n 1 n 1 + θ j(a j + 1) a j+1 >0 n 1 P n 1(a+e j e j+1 ) where given that a coalescence occurs and that the descendant sample has (, a j, a j+1, ), the ancestral one has (, a j + 1, a j+1 1, ) and the probability that one of the a j + 1 alleles with j gene copies is chosen to duplicate is j(a j + 1)/(n 1). Rappel : les effectifs de l échantilons (vecteur a est ordonné selon les effectifs des alleles = allele frequency spectrum = a j est le nombre de l alleles ayant j copies dans l échantilon) Griffiths and Tavaré : recursion for mutation models defined by a matrix (p ij ) of mutation rates from i to j

54 The recursion of Griffiths et al. Coalescent-based likelihood at a given point of the parameter space is an integral over all possible histories (genealogies with mutations) H = {H k ; k = 0,..., τ} corresponding to all coalescent or mutation events that occurred from H 0 the current sample state to H τ the allelic state of the most recent common ancestor (MRCA) of the sample.

55 The recursion of Griffiths et al. Then for any given state H k of the history (cf. Ewens) : p(h k ) = p(h k H k )p(h k ) {H k } where H k is the ancestral sample state (i.e. the state before the last event) and p(h k H k ) are the forward transition probabilities (i.e. from the ancestral to the current state)

56 The recursion of Griffiths et al. Griffiths & Tavaré 1994 : example for a single population p(h k = η) = ( n(n 1) 2N 1 + nµ) (nµ i j n j >0,j i n(n 1) n j 1 + ( 2N j n j >1 n 1 p(h k = η e j)). - Setting θ = 4Nµ and β = n(n 1 + θ), we have p(h k = η) = 1 β θ i j n j >0,j i n i + 1 n p ijp(h k = η e j + e i )) (n i + 1)p ij p(h k = η e j + e i ) + n (n j 1)p(H k = η e j ), j n j >1

57 The recursion of Griffiths et al. Griffiths & Tavaré 1994 : example for a single population p(h k = η) = 1 β θ i j n j >0,j i (n i + 1)p ij p(h k = η e j + e i ) + n (n j 1)p(H k = η e j ), j n j >1 Such recursions are too difficult to solve except for very simple models (WF + IAM, cf Ewens) Griffiths & Tavaré (1994) proposed to use a Monte Carlo approach using sequential importance sampling on past histories to solve the recursion.

58 Griffiths et al. s IS Griffiths et al s recursion Old IS scheme New IS scheme A general method based on diffusion Approximations of π

59 Inference of the likelihood by simulation Griffiths & Tavaré 1994 : p(h k = η) = 1 β θ i or equivalently p(h k ) = w GT (H k )( j n j >0,j i (n i + 1)p ij p(h k = η e j + e i ) + n (n j 1)p(H k = η e j ), j n j >1 i,j n j >0,j i M ij (H k )p(h k e j + e ai ) + C j (H k )p(h k e j )) j n j >1

60 Inference of the likelihood by simulation Griffiths & Tavaré 1994 : Backward absorbing Markov chain based on forward transition probabilities p(h k ) = w GT (H k )( i,j n j >0,j i M ij (H k )p(h k e j + e ai ) + C j (H k )p(h k e j )) j n j >1 Histories are build backward event by event using absorbing Markov chain (abs. state = MRCA) based on forward transitions probabilities ( uniform sampling based on M ij (H k ) and C j (H k )) among all possible events. w GT (H k ) is the weight associated with the IS proposal.

61 Inference of the likelihood by simulation Expending the recursion p(h k ) = {Hk } p(h k H k )p(h k ) over all possible ancestral histories of a current sample leads to Then p(h 0 ) = E [p(h 0 H 1 )...p(h τ 1 H τ )p(h τ )] L(P; D) = p(h 0 ) = H W GT (H)f GT (H) 1 L L τ 1 w GT ((H h ) k ). L h=1 k=0 L h=1 W GT (H h ) This IS scheme f GT (H) is not very efficient because it does not appropriately consider that some backward transitions are more likely than others given the current state (example : SMM mutation).

62 Griffiths et al. s IS Griffiths et al s recursion Old IS scheme New IS scheme A general method based on diffusion Approximations of π

63 Towards a better IS scheme (Stephens & Donnelly 2000, de Iorio & Griffiths 2004) A better Importance Sampling (IS) scheme should be used : Let Q(H k ) be a given distribution, then p(h k H k ) p(h k ) = {H k } Q(H k ) Q(H k )p(h k ) = E Q [ p(h 0 H 1 ) Q(H 1 )...p(h τ 1 H τ ) ] Q(H τ ) where E Q is expectation over the distribution of full histories induced by Q. This means that Q may be viewed as a proposal distribution in a sequential IS algorithm with matching weights p(h k H k /Q(H k )).

64 Towards a better IS scheme (Stephens & Donnelly 2000, de Iorio & Griffiths 2004) A better Importance Sampling (IS) scheme should be used : Let Q(H k ) be a given distribution, then p(h k ) = E Q [ p(h 0 H 1 ) Q(H 1 )...p(h τ 1 H τ ) ] Q(H τ ) where E Q is expectation over the distribution of full histories induced by Q. This means that Q may be viewed as a proposal distribution in a sequential IS algorithm with matching weights p(h k H k /Q(H k )). The problem is then to find the proposal distribution that minimizes the variance of likelihood estimates 1 L L τ w GT ((H h ) k ). h=1 k=0

65 Towards a better IS scheme (Stephens & Donnelly 2000, de Iorio & Griffiths 2004) The ideal proposal is the backward transition probability p(h k H k ), because the IS weights are then p(h k ) Q(H k ) = p(h k H k ) p(h k H k ) = p(h k) p(h k ) and thus their product is always the sample likelihood, p(h 0 ). expliciter a single tree reconstruction allows exact likelihood computations (null variance). However, backward transition probabilities p(h k H k ) are generally unknown Aim : find good approximations ˆp(H k H k ) of p(h k H k )

66 Towards a better IS scheme (Stephens & Donnelly 2000, de Iorio & Griffiths 2004) The likelihood at a given point is an integral over all possible histories H = {H k ; k = 0,..., τ}. Markov coalescent process p(h k ) = p(h k H k )p(h k ) and p(h 0 ) = E [p(h 0 H 1 )...p(h τ 1 H τ )p(h τ )]. However, forward transition probabilities p(h k H k ) are not efficient in a backward process Importance sampling techniques based on an approximation ˆp(H k H k ) of p(h k H k ) are used to build more likely histories p(h 0 ) = Eˆp [ p(h 0 H 1 ) ˆp(H 1 H 0 )...p(h τ 1 H τ ) ˆp(H τ H τ 1 ) ].

67 Linking optimal weights to addition of a gene to a sample Represent sample probability p(n) as integral over the joint distribution f (x) of allele frequencies in the population p(n) = x p(n x)f (x) dx = E f [( n n ) i X n i i ] where is the binomial coefficient. ( n n ) = n! i n i!

68 Linking optimal weights to addition of a gene to a sample Represent sample probability p(n) as integral over the joint distribution f (x) of allele frequencies in the population p(n) = x p(n x)f (x) dx = E f [( n n ) i X n i i ] Then the joint probability that we have a sample n and that an additional gene copy is of type j is E f [X j ( n n ) i X n i i ] = n j + 1 n + 1 p(n + e j).

69 Linking optimal weights to addition of a gene to a sample Then the joint probability that we have a sample n and that an additional gene copy is of type j is E f [X j ( n n ) i X n i i ] = n j + 1 n + 1 p(n + e j). We write this joint probability as p(n) times π(j n), where π is thus the probability that an additional gene is of type j, given we have already drawn the sample n from the population. Thus if H k and H k differ by the addition of one gene of type j, we can write the optimal IS weight as p(h k = n) p(h k = n + e j ) = n j n + 1 π(j n)

70 Towards a better IS scheme : the π s Let π( H k ) be the conditional distribution of the allelic type of a n + 1 gene, given H k the configuration (i.e. allelic types) of the first n genes of the sample. Then the optimal IS distribution (exact backward transition probabilities) is, for a single population : p(h k H k ) = 1 β θn π(i H k e j ) j π(j H k e j ) P ij p(h k H k ) = 1 β n j (n j 1) π(j H k e j ) for H k = H k e j + e i for H k = H k e j

71 Towards a better IS scheme : the ˆπ s Unfortunately, π s are generally unknown Stephens & Donnelly (2000) proposed a good approximation ˆπ for the πs for a single WF population. de Iorio & Griffiths (2004) proposed a general method for appoximating the πs under different mutational and demographic models Then approximate backward transition probabilities using the ˆπs are used : ˆp(H k H k ) = 1 β θn ˆπ(i H k e j ) j ˆπ(j H k e j ) P ij ˆp(H k H k ) = 1 n j (n j 1) β ˆπ(j H k e j ) for H k = H k e j + e i for H k = H k e j

72 Griffiths et al. s IS Griffiths et al s recursion Old IS scheme New IS scheme A general method based on diffusion Approximations of π

73 The backward equation for f (X t X 0 = x) Pour un processus de diffusion, la densité de probabilité f des fréquences alléliques satisfait l équation arrière de Kolmogorov, qui décrit les changements de f au cours du temps sous la forme df (X t X 0 = x) dt = Φ(f (x)), où Φ est un opérateur différentiel qui prend ici la forme avec Φ = 1 2 i E j E = j E Φ j x i (δ ij x j ) x j 2 + ( x i r ij ) x i x j j E i E x j R = {r ij } θ 2 (P I ) où P = {p ij } est la matrice de mutation, et I la matrice identité.

74 The backward equation for E[g(X t ) X 0 = x] In the same way as df (X t X 0 = x) dt = Φ(f (x)), the following generator equation (Karlin and Taylor, 1981, p.215) holdsfor any function g(x) with bounded second derivatives E[g(X t ) X 0 = x] g(x) lim = Φ(g(x)). t 0 t We will apply this result with g the sample probability given population allele frequencies x.

75 ˆπ s computation Pour obtenir une récurrence sur les probabilités p(n) avec n = H 0 de l échantillon, on écrit p(n) sous la forme E [g(x)] p(n) = E[( n n ) i X n i i ] où ( n n ) = n! i n i!. On a donc d(p(n)) = Φ [p(n)]. dt A l équilibre stationnaire, d(p(n))/dt est nulle. En développant l expression pour Φ [p(n)], on retrouve alors la récurrence entre les p(n).

76 Explicit recursions in terms of π Applying the previous arguments then leads to a relation between probabilities of samples that differ by one coalescence or mutation event : N (n ( n 1 N + µ)) p(n) = N j n n j 1 N p(n e j) + Nµ j P ij (n i + 1 δ ij )p(n e j + e i ). i Expressing all p(.) in terms of p(n e j )s for distinct js : N ( n 1 j N + µ)π(j n e j)np(n e j ) = N n n j 1 d,j N p(n e j) + Nµ j P ij nπ(i n e j )p(n e j ) i...huge system of linear equations, not easier to solve in this form.

77 Griffiths et al. s IS Griffiths et al s recursion Old IS scheme New IS scheme A general method based on diffusion Approximations of π

78 ˆπ s computation On note que Φ [p(n)] peut s écrire sous la forme j E Φ j [p(n)], x j La technique d approximation développée par de Iorio & Griffiths est d approximer les π (dérivés précédemment de p(n), solution de Φ [p(n)] = 0) par des ˆπ dérivés des solutions de E[Φ j p(n) x j ]= E[Φ j x j ( n n ) i x n i i ]= 0, pour chaque j E.

79 ˆπ s computation La technique d approximation développée par de Iorio & Griffiths est d approximer les π (dérivés précédemment de p(n), solution de Φ [p(n)] = 0) par des ˆπ dérivés des solutions de E[Φ j ( n x j n ) i x n i i ]= 0, pour chaque j E. ce qui donne, pour une population panmictique, pour chaque j E n j (n 1 + θ)ˆp(n) = n(n j 1)ˆp(n e j ) + θp ij (n i + 1 δ ij )ˆp(n e j + e i ) i E

80 ˆπ s computation Rappel : π(j n) peut être exprimé en fonction de p(n) et p(n + e j ) : π(j n)p(n) = n j + 1 n + 1 p(n + e j). Si l on considère que cette relation est aussi valable pour les ˆπ et ˆp, ce qui ne sera généralement pas le cas, on a ˆπ(j n)ˆp(n) = n j + 1 n + 1 ˆp(n + e j)

81 ˆπ s computation Approximer les p(n), solutions de Φ [p(n)] = 0, par les ˆp(n) solutions de E[Φ j ( n x j n ) i x n i i ]= 0, pour chaque j E. ce qui donne, pour une population panmictique, pour chaque j E n j (n 1 + θ)ˆp(n) = n(n j 1)ˆp(n e j ) + θp ij (n i + 1 δ ij )ˆp(n e j + e i ) i E et en utilisant ˆπ(j n)ˆp(n) = n j +1 n+1 ˆp(n + e j) et remplaçant n par n + e j, on obtient donc pour chaque j E : (n 1 + θ)ˆπ(j n) = n j + θp ij ˆπ(i n) i E C est le systeme linéaire permettant le calcul des ˆπ(j n) pour un modèle de Wright-Fisher.

82 New IS scheme with the ˆπ s faire deux dipaos de bilan du nouveau schema d IS bilan en reprennant des bouts des diapos 63, 64, 65, 66, 70 et 71

83 A much better IS scheme based on the ˆπ s Drastic gain in efficiently with this new IS scheme (old IS : millions of trees) extract backward transition probabilities for a WF model with parent independent mutation (i.e. KAM) only 30 histories necessary for a good estimation of the likelihood for more complex models (structured populations & KAM) but efficiency slightly decrease with non parent-independent mutations models, e.g. stepwise mutation model (200 histories for structured populations & SMMM) and still limited efficiency for time inhomogeneous demographic models, e.g. one population with past size change (cf. Orang-Utan example) up to 20,000 histories necessary for strong disequilibrium scenarios (e.g. quick change in population size)

84 Implementations of IS : Genetree and Migraine Genetree (Bahlo & Griffiths 2000, old IS algorithm) - 2 to 4 populations with migration (ISM) Migraine (Rousset & Leblois , new IS algorithms) - One single stable population (KAM, SMM, GSM, ISM) - One pop. with past size variation (KAM, SMM, GSM, ISM) - 2 populations with migration (KAM, SMM, ISM) - Isolation By Distance in 1D and 2D (KAM)

85 Implementation of IS in Migraine 1. C++ core IS computations Stratified random sampling of parameter points Estimation of the likelihood at each point using IS 2. R code for post-treatment Likelihood surface interpolation by Kriging Inference of MLEs and CIs Plots of 1D and 2D likelihood profiles

86 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Griffiths et al. s IS Simulation tests Conclusions

87 Simulation tests Can we trust the demographic / historical inferences made with those methods?

88 Simulation tests Can we trust the demographic / historical inferences made with those methods? Aim Assess validity and robustness of the method : Bias, RMSE, coverage properties of confidence intervals robustness to realistic but uninteresting mis-specifications to this aim, we tested by simulation : - The performances of Migraine to infer dispersal under IBD - The performances of MsVar and Migraine to detect and measure past pop size changes

89 Simulation tests Can we trust the demographic / historical inferences made with those methods? Aim Assess validity and robustness of the method : Bias, RMSE, coverage properties of confidence intervals robustness to realistic but uninteresting mis-specifications to this aim, we tested by simulation : - The performances of Migraine to infer dispersal under IBD - The performances of MsVar and Migraine to detect and measure past pop size changes few interesting results...

90 Simulation tests Precision Validation Robustness MCMC vs. IS

91 Simulation tests (MsVar Girod et al. 2011) strong correlations between some pairs of natural parameters but this is expected given the coalescent theory...

92 Simulation tests (MsVar Girod et al. 2011) There is no information in the genetic data to infer µ, N and T separately because coalescent histories (H, genealogies with mutations) generated with the usual diffusion/coalescent approximations (large N, small µ) only depends on the scaled parameters Nµ and T /N constant Nµ product same unscaled history and same polymorphism Two indistinguishable situations under the coalescent approximations!

93 Simulation tests (MsVar Girod et al. 2011) Much better results by rescaling parameters as in the coalescent approximations

94 Simulation tests (Migraine) Nµ D 2N ancµ rel. bias & rel. RMSE Good reliability of the estimates for population declines, provided they are neither too recent, nor too weak BDR: FEDR: D Why does the method s performance strongly depend upon the time of the event, and its intensity?

95 Simulation tests (MsVar & Migraine) How genealogies are affected by demographic parameters? Predict the quantity of information present in the data The information in the data strongly depends on the number of mutations and coalecent events during the different demographic phases

96 Simulation tests Precision Validation Robustness MCMC vs. IS

97 Simulation tests (Migraine) Beyond biases, RMSE et bottleneck detection rates... ECDF of P values Nmu = 0.4 KS: Rel. bias, rel. RMSE 0.116, Nancmu = 400 KS: Rel. bias, rel. RMSE , D = 1.25 KS: Rel. bias, rel. RMSE , 0.14 Nratio = DR: 1 ( 0 ) KS: Rel. bias, rel. RMSE 0.152, (usually )GOOD

98 Simulation tests (Migraine) Beyond biases, RMSE et bottleneck detection rates... Testing CI coverage properties using LRT P-value distributions ECDF of P values Nmu = 0.4 KS: Rel. bias, rel. RMSE 0.116, Nancmu = 400 KS: Rel. bias, rel. RMSE , D = 1.25 KS: Rel. bias, rel. RMSE , 0.14 Nratio = DR: 1 ( 0 ) KS: Rel. bias, rel. RMSE 0.152, (usually )GOOD

99 Simulation tests (Migraine) Beyond biases, RMSE et bottleneck detection rates... Testing CI coverage properties using LRT P-value distributions ECDF of P values Nmu = 0.4 KS: Rel. bias, rel. RMSE 0.116, Nancmu = 400 KS: Rel. bias, rel. RMSE , D = 1.25 KS: Rel. bias, rel. RMSE , 0.14 Nratio = DR: 1 ( 0 ) KS: Rel. bias, rel. RMSE 0.152, Extremely recent and strong 10 Generations, D = N ratio = (θ anc = 400.0) (usually )GOOD (very rarely) BAD

100 Simulation tests Precision Validation Robustness MCMC vs. IS

101 Simulation tests (Migraine) Microsatellite markers show complex mutation processes Mutations do not fit SMM, indels of more than one repeat often occur

102 Simulation tests (Migraine) Microsatellite markers show complex mutation processes Mutations do not fit SMM, indels of more than one repeat often occur Better mutation model = Generalized Stepwise Model (GSM) indels of X (geometric) repeats at each mutation event commonly found value in natura : pgsm 0.22

103 Simulation tests (Migraine) Microsatellite markers show complex mutation processes Mutations do not fit SMM, indels of more than one repeat often occur Better mutation model = GSM indels of X (geometric) repeats commonly found value in natura : pgsm 0.22 Problem : Analyses under the SMM of data simulated under a GSM in a stable population often show false signs of bottleneck (57% of false detection with pgsm = 0.22)

104 Simulation tests Precision Validation Robustness MCMC vs. IS

105 Simulation tests (MsVar vs. Migraine) Some comparison with MsVar Similar performances for good scenarios Better bottleneck detection rate for non-optimal scenarios Parameter inference and CIs are slightly more accurate But comparison is not easy Frequentist vs. bayesian approaches very long computation times for MCMC (and sometimes for IS)

106 Conclusions from the simulation tests (MCMC & IS) Very efficient for bottleneck detections Accurate inferences for most demographic scenarios IS faster and sometimes more accurate than MCMC But : Not robutst to mutational processes Not robust to immigration (structured populations) Inaccurate for extremely strong and recent pop size change and... very long computation times for large data sets with many loci (i.e. NGS >> 100-1,000 loci)

107 Introduction Likelihoods under the coalescent Felsenstein et al. s MCMC Griffiths et al. s IS Simulation tests Conclusions

108 Conclusions Coalescent theory and ML-based approaches provide a powerful framework for statistical inference in population genetics. They extract much more information from the data than moment based methods. In these methods, gene genealogies are missing data Coalescent theory may also help understanding the limits of these methods (the reliability of a method also depends upon the quantity of information available in the data) Testing methods by simulation greatly helps to clearly understand real data analyses

109 Books

110 écrit pour pop subdiv, mais pas utile ici? p(n) = E[ ( n d (n d di ) ) i X n di di ]

111 écrit pour pop subdiv, mais pas utile ici? The backward diffusion equation holds with an operator which is a sum over different demes : Φ= 1 2 N demes d tot allele pairs i,j x N di (δ ij x dj ) 2 d x di x + d i M di dj x di where x di is the frequency of allele i in deme j.

112 écrit pour pop subdiv, mais pas utile ici? Applying the previous arguments then leads to a relation between probabilities of samples that differ by one coalescence/mutation/migration event : N tot ( n d ( n d 1 + m d + µ)) p(n) = d N d n dj 1 N tot n d p(n e dj ) d,j N d + N tot µ d,j P ij (n di + 1 δ ij )p(n e dj + e di ) i n d + N tot n d m j + 1 dd d,j d d n d + 1 p(n e dj + e d j).

113 écrit pour pop subdiv, mais pas utile ici? Expressing all p(.) in terms of p(n e dj )s for distinct d, j : N tot ( n d 1 + m d + µ)π(j d, n e dj )n d p(n e dj ) = d,j N d n dj 1 N tot n d p(n e dj ) d,j N d + N tot µ d,j P ij n d π(i d, n e dj )p(n e dj ) i + N tot n d m dd π(j d, n e dj )p(n e dj ) d,j d d

114 écrit pour pop subdiv, mais pas utile ici? Expressing all p(.) in terms of p(n e dj )s for distinct d, j : N tot ( n d 1 + m d + µ)π(j d, n e dj )n d p(n e dj ) = d,j N d n dj 1 N tot n d p(n e dj ) d,j N d + N tot µ d,j P ij n d π(i d, n e dj )p(n e dj ) i + N tot n d m dd π(j d, n e dj )p(n e dj ) d,j d d Huge system of linear equations, not easier to solve in this form.