Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Module de Master 2 Biostatistique: mod` eles de g´ en´ etique des populations
Likelihood-based demographic inference using the coalescent Rapha¨el Leblois & Fran¸cois Rousset Centre de Biologie pour la Gestion des populations (CBGP, UMR INRA)
Janvier 2014
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Introduction Likelihoods under the coalescent Felsenstein et al. MsVar Griffiths et al. Tests Conclusion
MsVar
Griffiths et al.
Tests
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 ?
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Methods based on coalescence simulations (Reminder...) Genealogy of the sample
forward in time
backward in time
Genealogy of the population
Coalescent tree
6
☇ ☇
?
;; P(Tk = t) ≈
k(k − 1) −t k(k−1) 2 e 2
P(m∣t) =
(µt)m e −µt m!
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
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
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
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
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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).
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 ∗ ∣Gk ) of observing the data D
given a specific genealogy Gk 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) = ∫ P(D; P∣G )P(G ; P)dG G
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Likelihood computations under the coalescent • The likelihood can be written as the sum of P(D; P∣Gk ) over
the genealogical space (all possible genealogies) : L(P; D) = ∫ P(D; P∣G )P(G ; P)dG G
Mutation
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
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Likelihood computations under the coalescent • The likelihood can be written as the sum of P(D; P∣Gk ) over
the genealogical space (all possible genealogies) : L(P; D) = ∫ P(D; P∣G )P(G ; P)dG G
...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) = EP(G ;P) (P(D; P∣G )) ≈
1 K ∑ P(D; P∣Gk ) K k=1
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Likelihood computations under the coalescent • The likelihood can be written as the sum of P(D; P∣Gk ) over
the genealogical space (all possible genealogies) : L(P; D) = ∫ P(D; P∣G )P(G ; P)dG G
...Usually impossible to sum over all possible genealogies...
→ Monte Carlo simulations are used : L(P; D) = EP(G ;P) (P(D; P∣G )) ≈
1 K ∑ P(D; P∣Gk ) K k=1
many many genealogies necessary for a good estimation of the likelihood...
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Likelihood computations under the coalescent
• Monte Carlo simulations are used :
L(P; D) = EP(G ;P) (P(D; P∣G )) ≈
1 K ∑ P(D; P∣Gk ) 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.
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 / contributing to ? 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) : 368–376.
IS Importance Sampling on genealogies, based on the work of Griffiths & Tavar´e 1994. Griffiths, R.C. and S. Tavar´ e (1994). Simulating probability distributions in the coalescent. Theor. Pop. Biol., 46 :131-159.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 &Tavar´e’s coalescent recursion - Extension to different models may be difficult - Implemented in fewer softwares (Genetree, Migraine)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 consider various models - Implemented in many softwares (LAMARC, Batwing, MsVar, MIGRATE, IM) IS Griffiths &Tavar´e’s coalescent recursion - Extension to different models may be difficult - Implemented in fewer softwares (Genetree, Migraine)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
The approach of Felsenstein et al. • Based on (1) on the availability of approximate exponential
distribution of coalescence (and migration and recombinaison) times and ; often, (2) on the separation of demographic and mutational processes : - First, the probability of a genealogy given the parameters of the demographic model P(Gk ; Pdemo ) can be computed from the distributions of time intervals between events. - Then the probability of the data given a genealogy and mutational parameters P(D; Pmut ∣Gk ) can be easily computed from the mutation model parameters, the mutation rate, tree topology and branch lengths of the tree. • From this, an efficient algorithm to explore the genealogical
and the parameter spaces should allow the inference of the likelihood over the two spaces.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
MCMC scheme 1. Init. Start with any scaled tree with mutations compatible with the data 2. Propose new “state” : parameter values and/or new genealogy 3. Accept or reject the new state 4. restart from 2. until “the chain converges to its stationary distribution” (N iterations) MCMC → correlated samples 5. End Discard B first iterations (burn-in) and sample every T iterations (thining), and use the remaining (N − B)/T samples to compute the likelihood/full posterior distribution
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
The Metropolis-Hastings (MH) algorithm (1953 - 1970)
The MH algorithm is a MCMC method for obtaining random samples from a probability distribution f (x) for which direct sampling is difficult.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
The Metropolis-Hastings (MH) algorithm (1953 - 1970) Basic description :
f (x) : desired distribution, g (x) ∝ f (x), Q(x∣y ) : transition
probabilities
1.Init. Choose an arbitrary point x0 2. Propose a candidate x ′ from Q(x ′ ∣xt ) 3. compute the acceptance ratio α = g (x ′ )/g (xt ) = f (x ′ )/f (xt ) If(α > 1), set xt+1 = x ′ otherwise with probability α set xt+1 = x ′ , and with probability (1 − α) set xt+1 = xt 4. restart from 2. until “the chain converges to its stationary distribution” 5. End burn-in, thining and use the remaining (N − B)/T samples to compute the likelihood/full posterior distribution
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Metropolis-Hastings sampling
For the Metropolis-Hastings algorithm, we need to compute the ratio of the probability of proposed update over the current state : 1. Computation of P(Gk ; Pdemo ) :
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
Metropolis-Hastings sampling
For the Metropolis-Hastings algorithm, we need to compute the ratio of the probability of proposed update over the current state : 1. Computation of P(Gk ; Pdemo ) : a. The conditional probability of occurrence of an event at ti+1 , given ti the time of the previous event and γ(t) the rate of events, is : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt) ti
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
Metropolis-Hastings sampling For the Metropolis-Hastings algorithm, we need to compute the ratio of the probability of proposed update over the current state : 1. Computation of P(Gk ; Pdemo ) : a. The conditional probability of occurrence of an event at ti+1 , given ti the time of the previous event and γ(t) the rate of events, is : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt) ti
→ The rate of events is the sum of the rates of occurrence of all potential events at time t, ex. with coalescences and migration : npop ⎛ kp (kp − 1) ⎞ + ∑ kq mp→q 4Np ⎠ p=1 ⎝ q=1,q≠p
npop
γ(t) = ∑
Np size of subpopulation p, mp→q migration rate from subpop p to q, kp number of lineages in subpop p
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
Metropolis-Hastings sampling For the Metropolis-Hastings algorithm, we need to compute the ratio of the probability of proposed update over the current state : 1. Computation of P(Gk ; Pdemo ) : a. The conditional probability of occurrence of an event at ti+1 , given ti the time of the previous event and γ(t) the rate of events, is : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt) ti
b. Then to compute P(Gk ; Pdemo ), we multiply over all events of the tree MRCA
MRCA
τ =1
τ =1
P(Gk ; Pdemo ) = ∏ P(τ ∣τ −1) = ∏ γ(tτ +1 )exp(− ∫
tτ tτ −1
γ(t)dt)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Metropolis-Hastings sampling For the Metropolis-Hastings algorithm, we need to compute the ratio of the probability of proposed update over the current state : 1. Computation of P(Gk ; Pdemo ) : MRCA
tτ
τ =1
tτ −1
P(Gk ; Pdemo ) = ∏ γ(tτ )exp(− ∫
γ(t)dt)
- Example for a WF population (coalescence only) MRCA
P(Gk ; Pdemo ) = ∏
τ =1
kτ (kτ − 1) −(tτ −tτ −1 ) kτ (k2τ −1) e 2
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Metropolis-Hastings sampling • old... First, we compute the conditional probability
of a demographic event given γ(t) the rate of events, as : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt)
ti
- Then to compute P(Gk ; Pdemo ), we multiply over all events of the tree MRCA
P(Gk ; Pdemo ) = ∏ P(τ ∣τ − 1) τ =1
faire arbre avec coalescences + migrations + intervalles de temps
MRCA
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
Metropolis-Hastings sampling • old... First, we compute the conditional probability of a demographic
event given γ(t) the rate of events, as : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt)
ti
• Then to compute P(Gk ; Pdemo ), we multiply over all events of the tree
P(Gk ; Pdemo ) =
TMRCA
∏ P(τ ∣τ − 1)
τ =1
- Example for a WF population (coalescence only) MRCA
P(Gk ; Pdemo ) = ∏
τ =1
kτ (kτ − 1) −(tτ −tτ −1 ) kτ (kτ −1) 2 e 2
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Metropolis-Hastings sampling 1. Computation of P(Gk ; Pdemo ) : MRCA
tτ
τ =1
tτ −1
P(Gk ; Pdemo ) = ∏ γ(tτ )exp(− ∫
γ(t)dt)
2 Then compute the probability P(D; Pmut ∣Gk ) of the data D given the genealogy Gk , by going from the MRCA to the leaves and considering the probability of occurrence of all mutations on each branch of length tb and their effects (i.e.transition among genetic states x → y ) :
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
Metropolis-Hastings sampling 2 Then compute the probability P(D; Pmut ∣Gk ) :
effect of mutations
P(D; Pmut ∣Gk ) =
nb branch
∏
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ P(y ∣x, mb ) ⋅
number of mutations
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ P(mb ∣tb )
b=1 2(n−1)
= ∏ ((Matmut )mb )x,y b=1
Mutation matrix : transition probability between genetic states (x, y )
(µtb )mb e −µtb mb !
Poisson probability for the mb mutations
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Metropolis-Hastings sampling 1. Computation of P(Gk ; Pdemo ) : MRCA
tτ
τ =1
tτ −1
P(Gk ; Pdemo ) = ∏ γ(tτ )exp(− ∫
γ(t)dt)
2 Then compute P(D; Pmut ∣Gk ) : 2(n−1)
P(D; Pmut ∣Gk ) = ∏ ((Matmut )mb )x,y b=1
(µtb )mb e −µtb mb !
3 These probabilities are plugged into the MH formula for acceptance probabilities of candidate changes for the next state of the Markov chain. Reminder : L(P; D) = EP(G ;P) (P(D; P∣G )) ≈
1 I
K
∑k=1 P(D; Pmut ∣Gk )P(Gk ; Pdemo )
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Coalescent-based MCMC example : MsVar
• One example of a coalescent-based MCMC algorithm : MsVar Beaumont, M. 1999. Detecting Population Expansion and Decline Using Microsatellites. Genetics.
• Biological contexte : Past changes in population sizes (cf.
Orang-Utans) - Details of the MCMC algorithm - few results on the Orang-Utan data set
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Coalescent-based MCMC example : MsVar • Demographic model : a single isolated panmictic (WF)
population with a exponential past change in population size.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 , Nanc + 1 mutation parameter µ 3 scaled parameters (diffusion approx.) : θ, D, θanc ,
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Coalescent-based MCMC example : MsVar • Mutation model : Stepwise Mutation Model (SMM)
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Coalescent-based MCMC example : MsVar
P = N, T, Nanc , µ P′ = θ, D, θanc
• Aim : infer those parameters (P or P ′ ) from a unique actual
genetic sample using coalescent-based MCMC algorithms
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
MH/MCMC of MsVar • 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 - and the parameter space
all algorithms based on the ’Felsenstein et al.’ approach uses similar MH/MCMC algorithms with slight differences in the MCMC update steps.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
MH/MCMC of MsVar To sample into the posterior distribution, P(D∣P), we need to compute the probability of the data for a given genealogy and given parameter values : P(D∣H, P) where H represents the genealogical and mutational history
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
MH/MCMC of MsVar To sample into the posterior distribution, P(D∣P), we need to compute the probability of the data for a given genealogy and given parameter values : P(D∣H, P) where H represents the genealogical and mutational history In the standard coalescent, all the lineages have the same probability to coalesce and mutate ; we can therefore reduce the history (genealogy and mutations) to a sequence of dated events i.e. the likelihood only depend upon the waiting times between events, not upon the topology itself. Credits : Claire Calmet’s PhD thesis (http ://tel.archives-ouvertes.fr/tel-00288526/en/)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
MH/MCMC of MsVar To sample into the posterior distribution, P(D∣P), we need to compute the probability of the data for a given genealogy and given parameter values : P(D∣H, P) where H represents the genealogical and mutational history 1. we compute the conditional probability of occurrence of an event at ti+1 , given an event at ti , as : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt)
ti
γ(t) =
⎛ k(k − 1) kθ ⎞ Nact λ(t) + , where λ(t) = 2 2⎠ N(t) ⎝
Credits : Claire Calmet’s PhD thesis (http ://tel.archives-ouvertes.fr/tel-00288526/en/)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
MH/MCMC of MsVar To sample into the posterior distribution, P(D∣P), we need to compute the probability of the data for a given genealogy and given parameter values : P(D∣H, P) where H represents the genealogical and mutational history 1. we compute the conditional probability of occurrence of an event at ti+1 , given an event at ti , as : P(ti+1 ∣ti ) = γ(ti+1 )exp(− ∫
ti+1
γ(t)dt)
ti
where γ(t) is the rate of the events (sum of the rates of occurrence of coalescences and mutations at t). 2. Then we multiply over all events of the sequence. Credits : Claire Calmet’s PhD thesis (http ://tel.archives-ouvertes.fr/tel-00288526/en/)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 (Nact , Nanc ), time of the event (T ) or mutation rate(µ). or Update the genealogie
both updates made using the Metropolis-Hastings algorithm
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
updates of genealogical histories Add or remove 2 mutations
Merge or split 1/2 mutation(s)
Change the order of 2 events
Change the ancestral lineages Add or remove 3 mutations
Credits : Claire Calmet’s PhD thesis (http ://tel.archives-ouvertes.fr/tel-00288526/en/)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
MCMC updates in MsVar T = times of events, r = population size ratio
M. Beaumont : “This scheme was devised by trial and error to obtain good rates of convergence.”
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
MCMC updates in MsVar
• for each update, the new state (P ′ or H ′ ) 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) rMH = •
P(D∣P ′ , H ′ )Prior(P ′ ) P(P ′ → P) P(D∣P, H)Prior(P) P(P → P ′ )
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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...cf. FR : Geyer arguments → 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 despite no clear convergence indices...
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Analyses of MsVar results
• Bayesian method → compare posteriors (plain) and priors
(dashed)
... and test different priors
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Analyses of MsVar results • Bayesian method → compute Bayes factor to check for
contraction or expansion signal BF =
(Posterior prob. model 1) (Prior prob. model 2) (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(Nanc /Nact > 1) Posterior P(Nanc /Nact < 1)
BF =
# MCMC steps where (Nanc /Nact > 1) # MCMC steps where (Nanc /Nact < 1)
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
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. 2006 PLoS Biology)
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
An application of MsVar : Orang-Utans and the deforestation of Borneo • The data
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
An application of MsVar : Orang-Utans and the deforestation of Borneo • MsVar results
→ MsVar efficiently detects a past decrease in population size
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
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)
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 &Tavar´e’s coalescent recursion (cf. Ewens’ recursion) - Extension to different models may be difficult - Implemented in fewer softwares (Genetree, Migraine)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 = {Hk ; k = 0, −1, ..., −m} corresponding to all coalescent or mutation events that occurred from H0 the current sample state to H−m the allelic state of the most recent common ancestor (MRCA) of the sample.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
The recursion of Griffiths et al.
• Then for any given state Hk of the history (cf. Ewens) :
p(Hk ) = ∑ p(Hk ∣Hk−1 )p(Hk−1 ) {Hk−1 }
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
The recursion of Griffiths et al. • Griffiths & Tavar´ e 1994 : example for a single population p(Hk = η) =
⎡ ⎢ ⎢(nµ ∑ ∑ ni + 1 pij p(Hk−1 = η − ej + ei )) ⎢ n(n−1) na ( 2N + nµ) ⎢ i j∶nj >0,j≠i ⎣ ⎤ ⎥ n(n − 1) nj − 1 p(Hk−1 = η − eaj ))⎥ +( ∑ ⎥. 2N j∶nj >1 n − 1 ⎥ ⎦ 1
- Setting θ = 4Nµ and β = n(n − 1 + θ), we have ⎡ 1⎢ θ ∑ ∑ (ni + 1)pij p(Hk−1 = η − ej + ei ) p(Hk = η) = ⎢ β⎢ ⎢ i j∶nj >0,j≠i ⎣ ⎤ ⎥ + n ∑ (nj − 1)p(Hk−1 = η − ej )⎥ ⎥, ⎥ j∶nj >1 ⎦
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
The recursion of Griffiths et al. • Griffiths & Tavar´ e 1994 : example for a single population
- Setting θ = 4Nµ and β = n(n − 1 + θ), we have ⎡ 1⎢ p(Hk = η) = ⎢ θ ∑ ∑ (ni + 1)pij p(Hk−1 = η − ej + ei ) β⎢ ⎢ i j∶nj >0,j≠i ⎣ ⎤ ⎥ + n ∑ (nj − 1)p(Hk−1 = η − ej )⎥ ⎥, ⎥ j∶nj >1 ⎦
• Such recursions are too difficult to solve except for very simple
models (WF + IAM, cf Ewens) → Griffiths & Tavar´e (1994) proposed to use a Monte Carlo approach using importance sampling on past histories to solve the recursion.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Inference of the likelihood by simulation • Griffiths & Tavar´ e 1994 : ⎡ 1⎢ p(Hk = η) = ⎢ θ ∑ ∑ (ni + 1)pij p(Hk−1 = η − ej + ei ) β⎢ ⎢ i j∶nj >0,j≠i ⎣ ⎤ ⎥ + n ∑ (nj − 1)p(Hk−1 = η − ej )⎥ ⎥, ⎥ j∶nj >1 ⎦
or equivalently p(Hk ) = wGT (Hk )(
∑
i,j∶nj >0,j≠i
Mij (Hk )p(Hk − ej + eai )
+ ∑ Cj (Hk )p(Hk − ej )) j∶nj >1
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Inference of the likelihood by simulation • Griffiths & Tavar´ e 1994 :
Backward absorbing Markov chain based on forward transitions probabilities p(Hk ) = wGT (Hk )(
∑
i,j∶nj >0,j≠i
Mij (Hk )p(Hk − ej + eai )
+ ∑ Cj (Hk )p(Hk − ej )) j∶nj >1
→ Histories are build backward event by event using absorbing Markov chain (abs. state = MRCA) based on forward transitions probabilities (“uniform sampling” based on Mij (Hk ) and Cj (Hk )) among all possible events. wGT (Hk ) is the weight associated with the IS proposal.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Inference of the likelihood by simulation • Expending the recursion p(Hk ) = ∑{Hk−1 } p(Hk ∣Hk−1 )p(Hk−1 )
over all possible ancestral histories of a current sample leads to p(H0 ) = E [p(H0 ∣H−1 )...p(H−m+1 ∣H−m )p(H−m )] Then L(P; D) = p(H0 ) = ∫ WGT (H)fGT (H) ≈ H
≈
1 L ∑ WGT (Hh ) L h=1
1 L −m ∑ ∏ wGT ((Hh )k ). L h=1 k=0
This IS scheme fGT (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).
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Towards a better IS scheme (Stephens & Donnelly 2000, de Iorio & Griffiths 2004)
→ A better Importance Sampling (IS) scheme should be used : Let Q(Hk−1 ) be a new proposal distribution such that p(Hk ∣Hk−1 ) Q(Hk−1 )p(Hk−1 ) {Hk−1 } Q(Hk−1 )
p(Hk ) = ∑
= EQ [
p(H0 ∣H−1 ) p(H−m+1 ∣H−m ) ... ] Q(H−1 ) Q(H−m )
but need an efficient proposal distribution...
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Towards a better IS scheme (Stephens & Donnelly 2000, de Iorio & Griffiths 2004)
• The ideal proposal is the backward transition probability
p(Hk−1 ∣Hk ), then p(Hk ∣Hk−1 )
p(Hk−1 ) p(Hk ∩ Hk−1 ) = = p(Hk ) Q(Hk−1 ) p(Hk−1 ∣Hk )
→ a single tree reconstruction allows exact likelihood computations (null variance). • However, backward transition probabilities p(Hk−1 ∣Hk ) are
generally unknown Aim : find good approximations pˆ(Hk−1 ∣Hk ) of p(Hk−1 ∣Hk )
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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 = {Hk ; k = 0, −1, ..., −m}. • Markov coalescent process → p(Hk ) = ∑ p(Hk ∣Hk−1 )p(Hk−1 )
and p(H0 ) = E [p(H0 ∣H−1 )...p(H−m+1 ∣H−m )p(H−m )].
• However, forward transition probabilities p(Hk ∣Hk−1 ) are not
efficient in a backward process • Importance sampling techniques based on an approximation
pˆ(Hk−1 ∣Hk ) of p(Hk−1 ∣Hk ) are used to build more likely histories p(H0 ) = Epˆ [
p(H0 ∣H−1 ) p(H−m+1 ∣H−m ) ... ]. pˆ(H−1 ∣H0 ) pˆ(H−m ∣H−m+1 )
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Towards a better IS scheme : the π’s
• Let π(⋅∣Hk ) be the conditional distribution of the allelic type
of a n + 1 gene given Hk the configuration (i.e. allelic types) ’ of the first n genes of the sample.
• Then the optimal IS distribution f ∗ (exact backward
transition probabilities) is, for a single population : π(i∣Hk − ej ) 1 θnj Pij β π(j∣Hk − ej ) 1 nj (nj − 1) p(Hk−1 ∣Hk ) = β π(j∣Hk − ej )
p(Hk−1 ∣Hk ) =
for Hk−1 = Hk − ej + ei for Hk−1 = Hk − ej
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Griffiths et al.
MsVar
Tests
Conclusion
Towards a better IS scheme : the π ˆ ’s • Unfortunately, π’s are generally unknown → Stephens & Donnelly (2000) proposed a good approximation π ˆ for the π’s for a single WF population. → deIorio & Griffiths (2004) proposed a general method for computing the π ˆ ’s under different mutational and demographic models (solution of a linear system based on an approximation of the recursion, not detailed here) • Then approximate backward transition probabilities using the
π ˆ ’s are used : π ˆ (i∣Hk − ej ) 1 θnj Pij β π ˆ (j∣Hk − ej ) 1 nj (nj − 1) pˆ(Hk−1 ∣Hk ) = βπ ˆ (j∣Hk − ej )
pˆ(Hk−1 ∣Hk ) =
for Hk−1 = Hk − ej + ei for Hk−1 = Hk − ej
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
π ˆ ’s computation Pour un processus de diffusion, la densit´e de probabilit´e f des fr´equences all´eliques satisfait l’´equation arri`ere de Kolmogorov, qui d´ecrit les changements de f au cours du temps sous la forme df = Φ(f ), dt o` u Φ est un op´erateur diff´erentiel qui prend ici la forme ∂ 1 ∂2 Φ = ∑ ∑ xi (δij − xj ) + ∑ ( ∑ xi rij ) 2 i∈E j∈E ∂xi ∂xj j∈E i∈E ∂xj = ∑ Φj j∈E
avec
∂ ∂xj
θ R = {rij } ≡ (P − I ) 2 o` u P = {pij } est la matrice de mutation, et I la matrice identit´e.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
π ˆ ’s computation Pour obtenir une r´ecurrence sur les probabilit´e p(n) avec n = H0 de l’´echantillon, on ´ecrit p(n) sous la forme E [g (x)] n p(n) = E [( ) ∏ xini ] n i o` u n! n ( )= . n ∏i ni ! On a donc
d(p(n)) = Φ [p(n)] . dt A l’´equilibre stationnaire, d(p(n))/dt est nulle. En d´eveloppant l’expression pour Φ [p(n)], on retrouve alors la r´ecurrence entre les p(n).
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
π ˆ ’s computation On note que Φ [p(n)] peut s’´ecrire sous la forme ∑ Φj j∈E
∂ [p(n)] , ∂xj
La technique d’approximation d´evelopp´ee par deIorio & Griffiths est d’approximer les p(n), solutions de Φ [p(n)] = 0, par les pˆ(n) solutions de ∂p(n) ]= 0, pour tout j ∈ E , E [Φj ∂xj i.e. E [Φj
∂ n ( ) ∏ x ni ]= 0, pour tout j ∈ E . ∂xj n i i
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
π ˆ ’s computation La technique d’approximation d´evelopp´ee par deIorio & Griffiths est d’approximer les p(n), solutions de Φ [p(n)] = 0, par les pˆ(n) solutions de E [Φj
∂ n ( ) ∏ x ni ]= 0, pour tout j ∈ E . ∂xj n i i
ce qui donne, pour une population panmictique, pour tout j ∈ E nj (n − 1 + θ)ˆ p (n) = n(nj − 1)ˆ p (n − ej ) + ∑ θPij (ni + 1 − δij )ˆ p (n − ej + ei ) (1) i∈E
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
π ˆ ’s computation Toutes les permutations de l’ordre de tirage des g`enes de l’´echantillon sont ´equiprobables, en effet l’ordre des g`enes ´echantillonn´es n’est pas pris en compte dans les calcul de p(n). Cette notion d’´equiprobabilit´e des permutations des g`enes ´echantillonn´es implique la relation, dite relation de sym´etrie, suivante nj + 1 π(j∣n)p(n) = p(n + ej ). n+1 Si l’on consid`ere que cette relation de sym´etrie est aussi valable pour les π ˆ et pˆ, ce qui ne sera g´en´eralement pas le cas, on a π ˆ (j∣n)ˆ p (n) =
nj + 1 pˆ(n + ej ) n+1
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
π ˆ ’s computation
En int´egrant la relation de sym´etrie pour les pˆ et π ˆ dans le syst`eme d’´equation pr´ec´edent, on a pour tout a et pour tout j (n − 1 + θ) π ˆ (j∣n) = nj + ∑ θPij π ˆ (i∣n). i∈E
Ce syst`eme d’´equations donne th´eoriquement l’expression des π ˆ (⋅∣n). Ce syst`eme d’´equations est plus ou moins facile `a r´esoudre selon les mod`eles consid´er´es.
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
A much better IS scheme based on the π ˆ ’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)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Implementations of IS : Genetree and Migraine
• Genetree (Bahlo & Griffiths 2000, old IS algorithm) - 2 to 4 populations with migration (ISM) • Migraine (Rousset & Leblois 2007-2014, 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)
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
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
Tests
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Simulation tests Can we trust the demographic / historical inferences made with those methods ?
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
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
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
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...
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
Tests
(MsVar Girod et al. 2011)
strong correlations between some pairs of ”natural” parameters but this is expected given the coalescent theory . . .
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
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 !
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
Tests
Conclusion
(MsVar Girod et al. 2011)
Much better results by rescaling parameters as in the coalescent approximations
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests 20
_
_
rel. bias & rel. RMSE
Tests
Conclusion
(Migraine)
2N µ D 2N ancµ
_
_
_ 5 _ 2
_
_
1
_
_ _
_
_
BDR: 0.76 FEDR: 0
0.98 0
1 0
0.025 0.0625 0.125
_
_
_ __
_
_
_
__
1 0
1 0
1 0
1 0
0.98 0
0.79 0
0.5 0.005
0.25
0.5
1.25
2.5
3.5
5
7.5
_
Good reliability of the estimates for population declines, provided they are neither too recent, nor too weak. . .
_
__
_
0.2 -0.2
Griffiths et al.
_
_
10
0.5
MsVar
D
Why does the method’s performance strongly depend upon the time of the event, and its intensity ?
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
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
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
(Migraine)
Beyond biases, RMSE et bottleneck detection rates...
● ● ● ●●● ● ● ●● ● ●● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ●● ● ●● ● ●● ●● ● ●● ●●● ● ● ● ● ●● ●
0.4
1.0 0.8 0.6 0.4 0.0
0.6
0.8
Rel. bias, rel. RMSE 0.0496, 0.375
0.2
1.0
●● ● ● ● ●● ● ● ●●
KS: 0.433 0.4
0.6
0.8
1.0
Rel. bias, rel. RMSE −0.00452, 0.14
c(0, 1)
● ●● ●● ●● ●● ● ●● ●● ●● ● ● ●● ●● ●● ● ●● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ●● ● ●● ● ●● ●● ● ● ●
KS: 0.857
0.2
●●● ● ● ●● ●● ●● ●
0.0
Nratio = 0.001
0.2
0.8
1.0
2Nancmu = 400
0.6 0.4 0.2
1.0
1.0
0.8
●● ●● ●● ● ● ● ● ● ●● ●● ● ● ● ●● ●● ● ● ● ●● ● ●
● ●● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ●● ● ● ● ● ● ●● ●● ●
0.8
0.6
Rel. bias, rel. RMSE 0.116, 0.453
0.0
0.2 0.0
KS: 0.203 0.4
D = 1.25
0.6
0.2
● ● ● ●● ● ● ●
0.4
0.0
●● ● ● ● ●● ● ●● ● ● ●● ●● ● ●● ● ● ● ●● ●● ● ● ●● ●● ●●● ● ●● ●● ● ●● ● ● ●● ● ●● ● ●
●● ●● ● ● ●●● ●● ● ● ●● ● ● ● ● ●● ● ●
c(0, 1)
0.0
● ●● ● ● ● ●● ●● ●● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ●● ● ●
0.0 c(0, 1)
c(0, 1)
1) ECDF of c(0, P−values
0.2
0.4
0.6
0.8
1.0
2Nmu = 0.4
● ● ●● ●● ●
0.0
● ●● ● ● ● ● ● ●● ● ● ● ●● ●● ●● ● ●● ●● ● ● ●● ● ● ● ●● ● ● ●●
0.2
● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ●
●● ● ●● ● ● ● ● ● ● ● ●● ●● ●● ●
● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●● ● ● ●●
DR: 1 ( 0 ) KS: 0.165
0.4
0.6
0.8
Rel. bias, rel. RMSE 0.152, 0.601
(usually )GOOD
1.0
Tests
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
Tests
(Migraine)
Beyond biases, RMSE et bottleneck detection rates... Testing CI coverage properties using LRT P-value distributions 1.0 0.8 0.6 0.4 c(0, 1)
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KS: 0.857
0.4
0.6
0.8
Rel. bias, rel. RMSE 0.0496, 0.375
1.0
0.2
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KS: 0.433 0.4
0.6
0.8
1.0
Rel. bias, rel. RMSE −0.00452, 0.14
Nratio = 0.001
1.0
2Nancmu = 400
0.2
0.0
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0.8
1.0 0.8 0.6 0.4 0.2
1.0
0.6
0.8
●●● ● ● ●● ●● ●● ●
0.4
0.6
Rel. bias, rel. RMSE 0.116, 0.453
0.0
0.2 0.0
KS: 0.203 0.4
D = 1.25
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DR: 1 ( 0 )
0.2
0.2
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● ● ● ●● ● ● ●
0.0
0.0
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c(0, 1)
0.0
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0.0 c(0, 1)
c(0, 1)
1) ECDF of c(0, P−values
0.2
0.4
0.6
0.8
1.0
2Nmu = 0.4
KS: 0.165
● ● ●● ●● ●
0.0
0.2
0.4
0.6
0.8
Rel. bias, rel. RMSE 0.152, 0.601
(usually )GOOD
1.0
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
Tests
Conclusion
(Migraine)
Beyond biases, RMSE et bottleneck detection rates... Testing CI coverage properties using LRT P-value distributions 1.0 0.8 0.6 0.4 c(0, 1)
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KS: 0.857
0.4
0.6
0.8
Rel. bias, rel. RMSE 0.0496, 0.375
1.0
0.2
KS: 0.433 0.4
0.6
0.8
Extremely recent and strong 10 Generations, D = 0.025 Nratio = 0.001 (θanc = 400.0)
1.0
Rel. bias, rel. RMSE −0.00452, 0.14
Nratio = 0.001
1.0
2Nancmu = 400
0.2
0.0
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0.8
1.0 0.8 0.6 0.4 0.2
1.0
0.6
0.8
●●● ● ● ●● ●● ●● ●
0.4
0.6
Rel. bias, rel. RMSE 0.116, 0.453
0.0
0.2 0.0
KS: 0.203 0.4
D = 1.25
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DR: 1 ( 0 )
0.2
0.2
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0.0
0.0
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c(0, 1)
0.0
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0.0 c(0, 1)
c(0, 1)
1) ECDF of c(0, P−values
0.2
0.4
0.6
0.8
1.0
2Nmu = 0.4
KS: 0.165
● ● ●● ●● ●
0.0
0.2
0.4
0.6
0.8
1.0
Rel. bias, rel. RMSE 0.152, 0.601
(usually )GOOD
(very rarely) BAD
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
(Migraine)
Microsatellite markers show complex mutation processes • Mutations do not fit SMM,
indels of more than one repeat often occur
Tests
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
Tests
Conclusion
(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
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
MsVar
Griffiths et al.
(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 signs of bottleneck (57% of false detection with pGSM = 0.22)
Tests
Conclusion
Introduction
Likelihoods under the coalescent
Felsenstein et al.
Simulation tests
_
1
_ _
2
3
4
1e−03
_
_ _
_
_
5
_
_
1
2
3
1e+05
_
4
_ _
_ _
FEDR
2/5
2
1/5
0
0
_
_
1
_ _
_
_
_
1e−07
_
1
2
3
4
70 _
_
_
_ _
_
1
_
_ _
5
_
_
_
3
4
2
3
4
_
_ _
40 50
_
_
5
_
_
_
_
_
_
_
5/5
30
_
_ _
_
_
_ _
_
TwoNancmu
_
_
_
_
0
0
_
_
_
5
5/5
_
_
_
1
2
3
4
100
_
_
_
_
_
D _ _
2
3
4
_
_
_
_
_
_
_
_
_ _
_
_
_
_
_
_
5
1
_
_
5/5
_ _
5/5
0
0
NC=2/5
_
_
_
_
_
_
_
2
3
4
5
1
2
3
4
_
_ _
_ _
200
_
_
_
_
_
_
_
D _
_
_ _ _
1.0
_
• Frequentist vs. bayesian approaches
5
_ _
_
_
_
_
2
3
4
5
1
_
_
_
2
3
_
_
_
_
5/5 _
_ _
_
_
_ _
_
_
4
2
_ 1
_
_ _
5
_
_ _
_
50
_
_
TwoNancmu
_
_
2.0
_
10 20
_ _
5.0
_ _
0.5
TwoNmu
1e−07 1e−05 1e−03 1e−01 1e+01
_
_
5
D=1.25 (T=500 generations)
1
2
But comparison is not easy
_
_
_
D=0.25 (T=100 generations)
_
• Similar performances for “good” scenarios • Better bottleneck detection rate for “non-optimal” scenarios • Parameter inference seems more accurate
_
_
_
0.1
1e−07
_
1
_
_
0.2
_
0.5
_
_
50
_
TwoNancmu
_
20
_
10
_
2.0
_
1.0
1e+01 1e−03
TwoNmu
_
_ _
Some comparison with MsVar
5
D=0.125 (T=50 generations)
_
Conclusion
_
_ _
_
5
20
_
_
_
D
_
0.50 1.00
_
0.05 0.10 0.20
1e+01
_
_
1e−03
TwoNmu
_
Tests
(MsVarvs. Migraine)
D=0.025 (T=10 generations)
_
Griffiths et al.
_ _
_
1e+02
_
D _
_
TwoNancmu
_
_
_
_
_
_
1e−01
_
BDR
1e−04
_
_
_
1e+05
_
_
theta=0.4, Ancestral theta=40.0
1e−11
TwoNmu
1e−06 1e−03 1e+00 1e+03 1e+06
Migraine vs MsVar
MsVar
3
4
5
5/5
0
0
• very long computation times for MCMC
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
Conclusions from the simulation tests (MCMC & IS)
• Very efficient for bottleneck detections • Accurate inferences for most demographic scenarios • IS is faster and sometimes more accurate than the MCMC
equivalent But : • Not robutst to mutational processes • Not robust to immigration (structured populations) • Inaccurate for extremely strong and recent pop size change
Introduction
Likelihoods under the coalescent
Felsenstein et al.
MsVar
Griffiths et al.
Tests
Conclusion
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
