WITH the advent of ever more powerful computers

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1 Copyright Ó 2010 by the Genetic Society of America DOI: /genetic Bayeian Computation and Model Selection Without Likelihood Chritoph Leuenberger*,1,2 and Daniel Wegmann,1 *Département de Mathématique, Univerité de Fribourg, 1200 Fribourg, Switzerland and Computational and Molecular Population Genetic Laboratory, Intitute of Ecology and Evolution, Univerity of Bern, 3012 Bern, Switzerland Manucript received Augut 27, 2009 Accepted for publication September 2, 2009 ABSTRACT Until recently, the ue of Bayeian inference wa limited to a few cae becaue for many realitic probability model the likelihood function cannot be calculated analytically. The ituation changed with the advent of likelihood-free inference algorithm, often ubumed under the term approximate Bayeian computation (ABC). A key innovation wa the ue of a potampling regreion adjutment, allowing larger tolerance value and a uch hifting computation time to realitic order of magnitude. Here we propoe a reformulation of the regreion adjutment in term of a general linear model (GLM). Thi allow the integration into the ound theoretical framework of Bayeian tatitic and the ue of it method, including model election via Baye factor. We then apply the propoed methodology to the quetion of population ubdiviion among wetern chimpanzee, Pan troglodyte veru. WITH the advent of ever more powerful computer and the refinement of algorithm like MCMC or Gibb ampling, Bayeian tatitic have become an important tool for cientific inference during the pat two decade. Conider a model M creating data D (DNA equence data, for example) determined by parameter u from ome (bounded) parameter pace P R m whoe joint prior denity we denote by puþ. The quantity of interet i the poterior ditribution of the parameter, which can be calculated by Baye rule a pu jdþ¼c f M D j uþpuþ; where f M D j uþ i the likelihood of the data and c ¼ Ð P f MD j uþpuþdu i a normalizing contant. Direct ue of thi formula, however, i often prevented by the fact that the likelihood function cannot be calculated analytically for many realitic probability model. In thee cae one i obliged to ue tochatic imulation. Tavaré et al. (1997) propoe a rejection ampling method for imulating a poterior random ample where the full data D are replaced by a ummary tatitic (like the number of egregating ite in their etting). Even if the tatitic doe not capture the full information contained in the data D, rejection ampling allow for the imulation of approximate poterior ditribution of the parameter in quetion (the caled mutation rate in their model). Thi approach wa extended to multiple-parameter model with multivariate ummary tatitic ¼ 1 ;... ; n Þ T by Wei and von Haeeler (1998). In their etting a candidate vector u of parameter i imulated from a prior ditribution and 1 Thee author contributed equally to thi work. 2 Correponding author: Département de Mathématique, Univerité de Fribourg, Fribourg, Switzerland. chritoph.leuenberger@unifr.ch i accepted if it correponding vector of ummary tatitic i ufficiently cloe to the oberved ummary tatitic ob with repect to ome metric in the pace of, i.e.,ifdit(, ob ), e for a fixed tolerance e. We uppoe that the likelihood f M j uþ of the full model i continuou and nonzero around ob. In practice the ummary tatitic are often dicrete but the range of value i large enough to be approximated by real number. The likelihood of the truncated model M e ob Þ obtained by thi acceptance rejection proce i given by 1 f e j uþ ¼Ind 2B e ob ÞÞ f M j uþ f M j uþd ; 1Þ B e where B e ¼B e ob Þ¼f 2 R n j dit; ob Þ, eg i the e-ball in the pace of ummary tatitic and Ind() i the indicator function. Oberve that f e j uþ degenerate to a (Dirac) point meaure centered at ob a e/0. If the parameter are generated from a prior puþ, then the ditribution of the parameter retained after the rejection proce outlined above i given by p e uþ ¼ puþ Ð B e f M j uþd ÐP puþ Ð B e f M j uþddu : 2Þ We call thi denity the truncated prior. Combining (1) and (2) we get pu j ob Þ¼ f M ob j uþpuþ Ð P f M ob j uþpuþdu ¼ f e ob j uþp e uþ ÐP f e ob j uþp e uþdu : 3Þ Thu the poterior ditribution of the parameter under the model M for ¼ ob given the prior puþ i exactly Genetic 184: ( January 2010)

2 244 C. Leuenberger and D. Wegmann equal to the poterior ditribution under the truncated model M e ob Þ given the truncated prior p e uþ.ifwecan etimate the truncated prior and make an educated gue for a parametric tatitical model of M e ( ob ), we arrive at a reaonable approximation of the poterior pu j ob Þ even if the likelihood of the full model M i unknown. It i to be expected that due to the localization proce the truncated model will exhibit a impler tructure than the full model M and thu be eaier to etimate. Etimating p e uþ i traightforward, at leat when the ummary tatitic can be ampled from M in a reaonable amount of time: Sample the parameter from the prior puþ, create their repective tatitic from M, and ave thoe parameter whoe tatitic lie in B e ob Þ in a lit P¼fu 1 ;...; u N g. The empirical ditribution of thee retained parameter yield an etimate of p e uþ. Ifthe tolerance e i mall, then one can aume that f M j uþ i cloe to ome (unknown) contant over the whole range of B e ob Þ. Under that aumption, Equation 3 how that pu j ob Þp e uþ. However, when the dimenion n of ummary tatitic i high (and for more complex model dimenion like n ¼ 50 are not unuual), the cure of dimenionality implie that the tolerance mut be choen rather large or ele the acceptance rate become prohibitively low. Thi, however, ditort the preciion of the approximation of the poterior ditribution by the truncated prior (ee Wegmann et al. 2009). Thi ituation can be partially alleviated by peeding up the ampling proce; uch method are ubumed under the term approximate Bayeian computation (ABC). Marjoram et al. (2003) develop a variant of the claical Metropoli Hating algorithm (termed ABC MCMC in Sion et al. 2007), which allow them to ample directly from the truncated prior p e uþ. InSion et al. (2007) a equential Monte Carlo ampler i propoed, requiring ubtantially le iteration than ABC MCMC. But even when uch method are applied, the aumption that f M j uþ i contant over the e-ball i a very rough one, indeed. To take into account the variation of f M j uþ within the e-ball, a potampling regreion adjutment (termed ABC-REG in the following) of the ample P of retained parameter i introduced in the important article by Beaumont et al. (2002). Baically, they potulate a (locally) linear dependence between the parameter u and their aociated ummary tatitic. More preciely, the (local) model they implicitly aume i of the form u ¼ M 1 m 0 1 e, where M i a matrix of regreion coefficient, m 0 a contant vector, and e arandomvector of zero mean. Computer imulation ugget that for many population model ABC REG yield poterior marginal denitie that have narrower highet poterior denity (HPD) region and are more cloely centered aroundthetrueparametervaluethantheempirical poterior denitie directly produced by ABC ampler (Wegmann et al. 2009). An attractive feature of ABC REG i that the poterior adjutment i performed directly on the imulated parameter, which make etimation of the marginal poterior of individual parameter particularly eay. The method can alo be extended to more complex, nonlinear model a demontrated, e.g., in Blum and Francoi (2009). In extreme ituation, however, ABC REG may yield poterior that are nonzero in parameter region where the prior actually vanih (ee Figure 1B for an illutration of thi phenomenon). Moreover, it i not clear how ABC REG could yield an etimate of the marginal denity of model M at ob, information that i ueful for model comparion. In contrat to ABC REG we treat the parameter u a exogenou and the ummary tatitic a endogenou variable and we tipulate for M e ob Þ a general linear model (GLM in the literature not to be confued with the generalized linear model that unfortunately hare the ame abbreviation). To be precie, we aume the ummary tatitic created by the truncated model likelihood f e j uþ to atify j u ¼ Cu 1 c 0 1 e; 4Þ where C i a n 3 m matrix of contant, c 0 an n 3 1 vector, and e a random vector with a multivariate normal ditribution of zero mean and covariance matrix S : e N0; S Þ: A GLM ha the advantage of taking into account not only the (local) linearity, but alo the trong correlation normally preent between the component of the ummary tatitic. Of coure, the model aumption (4) can never repreent the full truth ince it tatitic are in principle unbounded wherea the likelihood f e j uþ i upported on the e-ball around ob. But ince the multivariate Gauian will fall off rapidly in practice and not reach far out off the boundary of B e ob Þ, thi i a diadvantage we can live with. In particular, the ordinary leat quare (OLS) etimate outlined below implie that for e/0 the contant c 0 tend to ob wherea the deign matrix C and the covariance matrix S both vanih. Thi mean that in the limit of zero tolerance e ¼ 0 our model aumption yield the true poterior ditribution of M. THEORY In thi ection we decribe the above methodology referred to a ABC GLM in the following in more detail. The baic two-tep procedure of ABC GLM may be ummarized a follow. GLM1: Given a model M creating ummary tatitic and given a value of oberved ummary tatitic ob, create a ample of retained parameter u j ; j ¼ 1;...; N, with the aid of ome ABC ampler (rejection ampling, ABC MCMC, or ABC PRC) baed on a prior ditribution puþ and ome choice of the tolerance e. 0.

3 Bayeian Computation Without Likelihood 245 Figure 1. Comparion of rejection (A and D), ABC REG (B and E), and ABC GLM (C and F) poterior with thoe obtained from analytical likelihood calculation. We etimated the population mutation parameter u ¼ 4Nm of a panmictic population for different oberved number of egregating ite (ee text). Shade indicate the L 1 ditance between the inferred and the analytically calculated poterior. White correpond to an exact match (zero ditance) and darker gray hade indicate larger ditance. If the inferred poterior differ from the analytical more than the prior doe, quare are marked in black. The top row (A C) correpond to cae with a uniform prior u Unif([0.005, 10]) and the bottom row (D F) to cae with a dicontinuou prior u Unif½0:005; 3Š[½6; 10ŠÞ with gap. The tolerance e i given a the abolute ditance in number of egregating ite. Shown are average over 25 independent etimation. To have a fair comparion, we adjuted the moothing parameter (bandwidth) to get the bet reult for all approache. GLM2: Etimate the truncated model M e ob Þ a a general linear model and determine, on the bai of the ample u j, from the truncated prior p e uþ an approximation to the poterior pu j ob Þ according to Equation 3. Let u look more cloely at thee two tep. GLM1: ABC ampling: We refer the reader to Marjoram et al. (2003) and Sion et al. (2007) for detail concerning ABC algorithm and to Marjoram and Tavaré (2006) for a comprehenive review of computational method for genetic data analyi. In practice, the dimenion of the ummary tatitic i often reduced by a principal component analyi (PCA). PCA alo ha a certain decorrelation effect. A more ophiticated method of reducing the dimenion of ummary tatitic, baed on partial leat quare (PLS), i decribed in Wegmann et al. (2009). In a recent preprint, Vogl et al. (C. Vogl, C. Futchik and C. Schloetterer, unpublihed data) propoe a Box Cox-type tranformation of the ummary tatitic that make the likelihood cloe to multivariate Gauian. Thi tranformation might be epecially efficient in our context a we aume normality of the error term in our model aumption. To fix the notation, let P¼fu 1 ;... ; u N g be a ample of vector-valued parameter created by ome ABC algorithm imulating from ome prior puþ and S ¼ f 1 ;... ; N g be the ample of aociated ummary tatitic produced by the model M. Each parameter u j i an m-dimenional column vector u j ¼u j ;... ; u j m Þt and each ummary tatitic i an n-dimenional column vector j ¼ j 1;... ; j n Þt 2B e ob Þ. The ample P and S can thu be viewed a m 3 N and n 3 N matrice P and S, repectively. The empirical etimate of the truncated prior p e uþ i given by the dicrete ditribution that put a point ma of 1/N on each value u j 2P. We mooth out thi empirical ditribution by placing a harp Gauian peak over each parameter value u j. More preciely, we et p e uþ ¼ 1 N X N j¼1 where fu u j 1 ; S u Þ¼ j 2pS u j and fu u j ; S u Þ; S u ¼ diag 1 ;... ; m Þ ÞtS 1 e 1=2Þu uj u u uj Þ 1=2 5Þ i the covariance matrix of f that determine the width of the Gauian peak. The larger the number N of ampled parameter value i, the harper the peak can be choen to till get a rather mooth p e. If the parameter domain P i normalized to [0, 1] m,ay,then areaonablechoicei k ¼ 1/N. Otherwie, k hould be adapted to the parameter range of the parameter component u k. Too mall value of k will reult in wiggly poterior curve, and too large value might unduly mear out the curve. The bet advice i to run the calculation with everal choice for S u. If p e induce a correlation between parameter, a nondiagonal S u might be beneficial. In practice, however, the poterior etimate are mot enitive to the diagonal value of S u. GLM2: general linear model: A explained in the Introduction, we aume the truncated model M e ob Þ to be normal linear; i.e., the random vector atify (4). The covariance matrix S encapulate the

4 246 C. Leuenberger and D. Wegmann trong correlation normally preent between the component of the ummary tatitic. C, c 0,andS can be etimated by tandard multivariate regreion analyi (OLS) from the ample P, S created in tep GLM1. [Strictly peaking, one mut redo an ABC ample from uniform prior over P to get an unbiaed etimate of the GLM if the prior puþ i not uniform already. On the other hand, ordinary leat-quare etimator are quite inenitive to the prior influence. In practice, one can a well ue the ample P to do the etimate. We applied both etimation method to the toy model preented in the example from population genetic ection and found no ignificant difference between the etimated poterior. The ame hold true for the o-called feaible generalized leatquare (FGLS) etimator; ee Greene (2003). In thi two-tage algorithm the covariance matrix i firt etimated a in our etting but in a econd round the deign matrix C i newly etimated. When we applied FGLS to our toy model, we found a difference in the etimated matrice only after the eighth ignificant decimal. FGLS i a more efficient etimator only when the ample ize are relatively mall a i often the cae in economical data et but not in ABC ituation. In theory, both OLS and FGLS are conitent etimator but FGLS i more efficient.] To be pecific, et X ¼ (1... P t ), where 1 i an N 3 1vectorof1.C and c 0 are determined by the uual leat-quare etimator ĉ 0.. ĈÞ¼SXX t XÞ 1 ; and for S we have the etimate Ŝ ¼ 1 N m ˆR t ˆR; 6Þ where ˆR ¼ S t X ĉ.ĉþ 0 t are the reidual. The likelihood for thi model dropping the hat on the matrice to unburden the notation i given by f e j uþ ¼ j2ps j 1=2 e 1=2Þ Cu c 0Þ t S 1 Cu c 0 Þ : 7Þ An exhautive treatment of linear model in a Bayeian (econometric) context i given in Zellner book (Zellner 1971). Recall from (3) that for a prior puþ and an oberved ummary tatitic ob, the parameter poterior ditribution for our full model M i given by pu j ob Þ¼c f e ob j uþp e uþ; 8Þ where f e ob j uþ i the likelihood of the truncated model M e ob Þ given by (7) and p e uþ i the etimated (and moothed) truncated prior given by (5). Performing ome matrix algebra (ee appendix a), one can how that the poterior (8) i up to a multiplicative contant of the form P N i¼j exp 1 2 Q jþ, where Q j ¼u t j Þ t T 1 u t j Þ ob c 0 Þ t S 1 ob c 0 Þ u j Þ t S 1 u uj v j Þ t Tv j : Here T, t j, and v j are given by and t j ¼ Tv j, where T ¼C t S 1 C 1 S 1 u Þ 1 9Þ v j ¼ C t S 1 ob c 0 Þ 1 S 1 u uj : 10Þ From thi we get where pu j ob Þ } XN j¼1 cu j Þe 1=2Þu tj Þ t T 1 u t j Þ ; 11Þ cu j Þ¼exp 1 2 uj Þ t S 1 u uj v j Þ t Tv j Þ : 12Þ When the number of parameter exceed two, graphical viualization of the poterior ditribution become impractical and marginal ditribution mut be calculated. The marginal poterior denity of the parameter u k i defined by pu k j Þ ¼ pu j Þdu k ; R m 1 where integration i performed along all parameter except u k. Recall that the marginal ditribution of a multivariate normal Nm; SÞ with repect to the kth component i the univariate normal denity Nm k ; k;k Þ. Uing thi fact, it i not hard to how that the marginal poterior of parameter u k i given by pu k j ob Þ¼a XN j¼1! cu j Þexp u k tkþ j 2 ; 13Þ 2t k;k where t k,k i the kth diagonal element of the matrix T, t k j i the kth component of the vector t j, and cu j Þ i till determined according to (12). The normalizing contant a could, in principle, be determined analytically but i in practice more eaily recovered by a numerical integration. Strictly peaking, the integration hould be done only over the bounded parameter domain P and not over the whole of R m. But thi no longer allow for an analytic form of the marginal poterior ditribution. For

5 Bayeian Computation Without Likelihood 247 large value of N the diagonal element in the matrix S u can be choen o mall that the error i in any cae negligible. Model election: The principal difficulty of model election method in nonparametric etting i that it i nearly impoible to etimate the likelihood of M at ob due to the high dimenion of the ummary tatitic (cure of dimenionality); ee Beaumont (2007) for an approach baed on multinomial logit. Parametric model on the other hand lend themelve readily to model election via Baye factor. Given the model M, one mut determine the marginal denity f M ob Þ¼ f ob j uþpuþdu: P It i eay to check from (1) and (2) that f M ob Þ¼A e ob ; pþ f e ob j uþp e uþdu: Here A e ob ; pþ :¼ P P puþ f M j uþddu B e 14Þ i the acceptance rate p of the rejection proce. It can eaily be etimated with aid of ABC REJ: Sample parameter from the prior puþ create the correponding tatitic from M and count what fraction of the tatitic fall into the e-ball B e centered at ob. If we aume the underlying model of M e ob Þ to be our GLM, then the marginal denity of M at ob can be etimated a f M ob Þ¼ A e ob ; pþ X N N j2pdj 1=2 j¼1 e 1=2Þ ob m j Þ t D 1 ob m j Þ ; 15Þ where the um run over the parameter ample P¼fu 1 ;... ; u N g, and D ¼ S 1 CS u C t m j ¼ c 0 1 Cu j : For two model M A and M B with prior probabilitie p A and p B ¼ 1 p A, the Baye factor B AB in favor of model M A over model M B i B AB ¼ f M A ob Þ f MB ob Þ ; 16Þ where the marginal denitie f MA and f MB are calculated according to (15). The poterior probability of model M A i f M A j ob Þ¼ B ABp A B AB p A 1 p B : EXAMPLES FROM POPULATION GENETICS Toy model: In Figure 1 we preent the comparion of poterior obtained with rejection ampling, ABC REG and ABC GLM, with thoe determined analytically ( true poterior ). A a toy model we inferred the population mutation parameter u ¼ 4Nm of a panmictic population model from the number of egregating ite S of a ample of equence with 10,000 bp for different oberved value and tolerance level. Etimation are alway baed on 5000 imulation with dit(s, S ob ), e, and we report the average of 25 independent replication per data point. Etimation bia of the different approache wa aeed by computing the total variation ditance between the inferred poterior and the true one obtained from analytical calculation uing the likelihood function introduced by Watteron (1975). Recall that the L 1 -ditance of two denitie f(u) and g(u) i given by d 1 f ; g Þ¼ 1 jf uþ guþj du: 2 It i equal to 1 when f and g have dijoint upport and it vanihe when the function are identical. When we ued a uniform prior u Unif([0.005, 10]) (Figure 1, A C), both ABC REG and ABC GLM give comparable reult and improve the poterior etimation compared to the imple rejection algorithm except for very low tolerance value e where the rejection algorithm i expected to be very cloe to the true poterior. The average total variation ditance over all oberved data et and tolerance value e are 0.236, 0.130, and for the rejection algorithm, ABC REG, and ABC GLM, repectively. Note that perfect matche between the approximate and the true poterior are difficult to obtain becaue all approximate poterior depend on a moothing tep that may not give accurate reult cloe to border of their upport. However, when we ued a dicontinuou prior u Unif([0.005, 3] [ [6, 10]) with an admittedly extremely artificial gap in the middle, we oberved a quite ditinct pattern (Figure 1, D and E). One clearly recognize that poterior inferred with ABC REG are frequently miplaced and often even farther away from the true poterior (in total variation ditance) than the prior, epecially for cae where the likelihood of the oberved data i maximal within the gap. The reaon for thi i that in the regreion tep of ABC REG parameter value may eaily be hifted outide the prior upport. Thi behavior of ABC REG ha been oberved earlier (Beaumont et al. 2002; Etoup et al. 2004; Tallmon et al. 2004) and a an ad hoc olution

6 248 C. Leuenberger and D. Wegmann TABLE 1 Mean and tandard deviation of the L 1 ditance between inferred and expected poterior for randomly generated GLM with N P ¼ 3, N S ¼ 4 [prior N(0, ), 200 imulation] p a d 1 (p 0, p e ) d 1 (p 0, p REG ) d 1 (p 0, p GLM ) KS tatitic b Figure 2. Example poterior for uniform (A) and dicontinuou (B) prior. The model i the ame a in Figure 1. Poterior etimate uing ABC GLM and ABC REG for S ob ¼ 16 were baed on 5000 imulation with dit(s, S ob ), 10. ABC REG poterior were moothed with a bandwidth of 0.4, and the width of the Dirac peak in the ABC GLM approach wa et to Hamilton et al. (2006) propoed to tranform the parameter value prior to the regreion tep by a tranformation of the form y ¼ lntanx aþ= b aþþp=2þþ 1 Þ, where a and b are the lower and upper border of the prior upport interval. For more complex prior like the dicontinuou prior ued here thi tranformation may not work. ABC GLM i much le affected by the gap prior than ABC REG. The average total variation ditance over all oberved data et and tolerance value e are 0.221, 0.246, and for the rejection algorithm, ABC REG, and ABC GLM, repectively. Example poterior with S ob ¼ 16 baed on 5000 imulation with dit(s, S ob ), 10 are hown in Figure 2. The ucce of ABC GLM depend on how well a general linear model fit the truncated model M e ob Þ. Under the null hypothei that the fit i perfect the etimated reidual r j (ee Equation 6) are independently multivariate normally ditributed random vector. Hence the Mahalanobi ditance d j ¼ r t j S 1 r j x 2 n 17Þ follow a x 2 -ditribution with n degree of freedom. A a quantification of model aement we propoe to report the Kolmogorov Smirnov tet tatitic for the empirical ditribution of d j and the reference x 2 -ditribution. (Reporting P-value will be of little ue in practice ince the null hypothei doe never hold exactly and hence the P-value will become very mall due to the large ample ize.) When the ummary tatitic are created from a general linear model, the fit hould be optimal. Thi i indeed the cae a the imulation reult in Table 1 how. We performed 200 imulation of randomly created general linear model with m ¼ 3 parameter, n ¼ 4 ummary tatitic, and a multivariate normal prior. The oberved tatitic were alo created from the repective model. For each imulated oberved tatitic and a Acceptance rate a a fraction. b KS tatitic decribing the linear model fit (ee text). different acceptance rate p ¼ 1.00, 0.50, 0.10, 0.05, and 0.01 we calculated the approximate poterior ditribution p e, p REG, and p GLM for the rejection algorithm, ABC REG, and ABC GLM, repectively. A the prior i multivariate normal, the true poterior p 0 can be analytically determined. Table 1 contain the mean and tandard deviation over the 200 imulation of the total variation ditance of the approximate poterior to the true poterior p 0 a well a the mean and tandard deviation of the Kolmogorov Smirnov tet tatitic for the GLM model fit. A i expected, the model fit i perfect [i.e., the Kolmogorov Smirnov (KS) tatitic i cloe to 0] for acceptance rate p ¼ 1. A the acceptance rate become lower, the model fit deteriorate ince the truncated model of a GLM i no longer exactly a general linear model. The total variation ditance to the true poterior increae lightly a p get maller but the improved rejection poterior p e motly outbalance the poorer model fit. A i expected in thi ideal ituation, ABC GLM and ABC REG ubtantially improve the poterior etimation over the pure rejection prior. To tet the other extreme we performed 200 imulation for a nonlinear one-parameter model with uniformly rather than normally ditributed error term; the prior wa again a normal ditribution. (The detail of thi toy model are decribed in appendix b.) A Table 2 how, the GLM model fit i already poor for an acceptance rate of p ¼ 1.00 (KS tatitic 0.10) and further deteriorate a p decreae. Note that the approximate poterior p REG and p GLM are cloer to the true poterior in average than p e and that both adjutment method perform imilarly. A expected, the accuracy of the poterior increae with maller acceptance rate, depite the fact that the model fit within the e-ball decreae. Thi ugget that the rejection tep contribute ubtantially to the etimation accuracy, epecially when the true model i nonlinear. We hould mention that in 30% of the imulation both ABC GLM and ABC REG actually increaed the ditance to the true poterior in comparion to the rejection poterior p e. A a rule of thumb we ugget that poterior

7 TABLE 2 Mean and tandard deviation of the L 1 ditance between inferred and expected poterior for the uniform error model (ee APPENDIX B) with N P ¼ 1, N S ¼ 5 {prior N(0, 2 2 ), error Unif[ 10, 10], 200 imulation} p a d 1 (p 0, p e ) d 1 (p 0, p REG ) d 1 (p 0, p GLM ) KS tatitic b a Acceptance rate a a fraction. b KS tatitic decribing the linear model fit (ee text). Bayeian Computation Without Likelihood 249 adjutment obtained by ABC GLM or ABC REG hould not be truted without further validation if the Kolmogorov Smirnov tatitic for the GLM model fit exceed a value of, ay, In that cae linear model are not ufficiently flexible to account for effect like nonlinearity in the parameter and nonnormality and heterocedaticity in the error term. In the etting of ABC REG a wider cla of model i introduced in Blum and Francoi (2009), where machine-learning algorithm are applied for the parameter etimation. Whether thee extenion can be applied in our context remain to be een. The advantage of the general linear model i that etimation can be done with ordinary leat quare and the important quantitie like marginal poterior and marginal likelihood can be obtained analytically. For more complex model thee quantitie will probably be acceible only via numerical integration, Monte Carlo method, etc. Application to chimpanzee: In tandard taxonomie, chimpanzee, the cloet living relative of human, are claified into two pecie: the common chimpanzee (Pan troglodyte) and the bonobo (P. panicu). Both pecie are retricted to Africa and diverged 9 MYA (Won and Hey 2005; Becquet and Przeworki 2007). The common chimpanzee are further ubdivided into three large population or ubpecie on the bai of their eparation by geographic barrier. Among them, the wetern chimpanzee (P. troglodyte veru) form the mot remote group. Interetingly, recent multilocu tudie found conitent level of gene flow between the wetern and the central (P. t. troglodyte) chimpanzee (Won and Hey 2005; Becquet and Przeworki 2007). Nonethele, a recent tudy of 310 microatellite in 84 common chimpanzee upport a clear ditinction between the previouly labeled population (Becquet et al. 2007). Uing a PCA analyi, indication for ubtructure within the wetern chimpanzee wa found in the ame tudy. To demontrate the applicability of the model election given in the theory ection we contrat two different model of the wetern chimpanzee population Figure 3. Baye factor for the iland relative to the panmictic population model for different acceptance rate (logarithmic cale). For very low acceptance rate we oberve large fluctuation wherea the Baye factor i quite table for larger value. Note that A e # correpond to #500 imulation, too mall a ample ize for robut tatitical model etimation. with thi data et: a model of a ingle panmictic population with contant ize and a finite iland model of contant ize and contant migration among deme. While we etimated u ¼ 2N e m, prior were et on N e and m eparately with log 10 (N e ) Unif([3, 5]) and m N( , ) truncated on m 2 [10 4,10 3 ]. In the cae of the finite iland model, we had an additional prior n pop Unif([10, 100]) on the number of iland, and individual were attributed randomly to the different iland. We obtained genotype for all 50 individual reported to be of wetern chimpanzee origin from the tudy of Becquet et al. (2007), excluding captive-born hybrid. We checked the propoed (Becquet et al. 2007) mutation pattern for each individual locu, and all allele not matching the aumed tepwie mutation model were et a miing data. A total of 265 loci were ued, after removing the loci on the X and the Y chromoome a well a thoe being monomorphic among the wetern chimpanzee. All imulation were performed uing the oftware SIMCOAL2 (Laval and Excoffier 2004) and we reproduced the pattern of miing data oberved in the data et. Uing the oftware package Arlequin3.0 (Excoffier et al. 2005), we calculated two ummary tatitic on the data et: the average number of allele per locu, K, and F IS, the fixation index within the wetern chimpanzee. We performed a total of 100,000 imulation per model. In Figure 3 we report the Baye factor of the iland model according to (16) for different acceptance rate A e ; ee (14). While there i a large variation for very mall acceptance rate, the Baye factor tabilize for A e $ Note that A e # correpond to,500 imulation and that the ABC GLM approach, baed on a model etimation and a moothing tep, i expected to

8 250 C. Leuenberger and D. Wegmann produce poor reult ince the etimation of the model parameter i unreliable due to the mall ample ize. The good new i that the Baye factor i table over a large range of tolerance value. We may therefore afely reject the panmictic population model in favor of population ubdiviion among wetern chimpanzee with a Baye factor of B DISCUSSION Due to till increaing computational power it i nowaday poible to tackle etimation problem in a Bayeian framework for which analytical calculation of the likelihood i inhibited. In uch cae, approximate Bayeian computation i often the choice. A key innovation in peeding up uch algorithm wa the ue of a regreion adjutment, termed ABC REG in thi article, which ued the frequently preent linear relationhip between generated ummary tatitic and parameter of the model u in a neighborhood of the oberved ummary tatitic ob (Beaumont et al. 2002). The main advantage i that larger tolerance value e till allow u to extract reaonable information about the poterior ditribution pujþ and hence le imulation are required to etimate the poterior denity. Here we preent a new approach to etimate approximate poterior ditribution, termed ABC GLM, imilar in pirit to ABC REG, but with two major advantage: Firt, by uing a GLM to etimate the likelihood function, ABC GLM i alway conitent with the prior ditribution. Second, while we do not find the ABC GLM approach to ubtantially outperform ABC REG in tandard ituation, it i naturally embedded into a tandard Bayeian framework, which in turn allow the application of well-known Bayeian methodologie uch a model averaging or model election via Baye factor. Our imulation how that the rejection tep i epecially beneficial if the true model i nonlinear for both ABC approache. ABC GLM i further compatible with any type of ABC ampler, including likelihood-free MCMC (Marjoram et al. 2003) or population Monte Carlo (Beaumont et al. 2009). Alo, more complicated regreion regime taking nonlinearity or heterocedacity into account may be enviioned when the GLM i replaced by ome more complex model. A great advantage of the current GLM etting i it implicity, which render implementation in tandard tatitical package feaible. We howed the applicability of the model election procedure via Baye factor by oppoing two different model of population tructure among the wetern chimpanzee P. troglodyte veru. Our analyi trongly ugget population ubtructure within the wetern chimpanzee ince an iland model i ignificantly favored over a model of a panmictic population. While none of our imple model i thought to mimic the real etting exactly, we till believe that they capture the main characteritic of the demographic hitory influencing our ummary tatitic, namely the number of allele K and the fixation index F IS. While the oberved F IS of 2.6% ha been attributed to inbreeding previouly (Becquet et al. 2007), we propoe that uch value may eaily arie if diploid individual are ampled in a randomly cattered way over a large, ubtructured population. While it wa almot impoible to imulate the value F IS ¼ 2.6% in the model of a panmictic population, it eaily fall within the range of value obtained from an iland model. We are grateful to Laurent Excoffier, David J. Balding, Chritian P. Robert, and the anonymou referee for their ueful comment on a firt draft of thi manucript. Thi work ha been upported by grant no. 3100A from the Swi National Foundation to Laurent Excoffier. LITERATURE CITED Beaumont, M., 2007 Simulation, Genetic, and Human Prehitory A Focu on Iland. McDonald Intitute Monograph, Univerity of Cambridge, Cambridge, UK. Beaumont, M., W. Zhang and D. Balding, 2002 Approximate Bayeian computation in population genetic. Genetic 162: Beaumont, M., R. C. Cornuet and J.-M. Marin, 2009 Adaptive approximate Bayeian computation. Biometrika (in pre). Becquet, C., and M. Przeworki, 2007 A new approach to etimate parameter of peciation model with application to ape. Genome Re. 17: Becquet, C., N. Patteron, A. Stone, M. Przeworki and D. Reich, 2007 Genetic tructure of chimpanzee population. Genome Re. 17: Blum, M., ando. Francoi, 2009 Non-linear regreion model for approximate Bayeian computation. Stat. Comput. (in pre). Etoup, A., M. Beaumont, F. Sennedot, C. Moritz and J. M. Cornuet, 2004 Genetic analyi of complex demographic cenario: patially expanding population of the cane toad, Bufo marinu. Evolution 58: Excoffier, L., G. Laval and S. Schneider, 2005 Arlequin (verion 3.0): an integrated oftware package for population genetic data analyi. Evol. Bioinform. Online 1: Greene, W., 2003 Econometric Analyi, Ed. 5. Pearon Education, Upper Saddle River, NJ. Hamilton, G., M. Stoneking and L. Excoffier, 2006 Molecular analyi reveal tighter ocial regulation of immigration in patrilocal population than in matrilocal population. Proc. Natl. Acad. Sci. USA 102: Laval, G., and L. Excoffier, 2004 Simcoal 2.0: a program to imulate genomic diverity over large recombining region in a ubdivided population with a complex hitory. Bioinformatic 20: Lindley, D., and A. Smith, 1972 Baye etimate for the linear model. J. R. Stat. Soc. B 34: Marjoram, P., and S. Tavaré, 2006 Modern computational approache for analying molecular genetic variation data. Nat. Rev. Genet. 10: Marjoram, P., J. Molitor, V. Plagnol and S. Tavaré, 2003 Markov chain Monte Carlo without likelihood. Proc. Natl. Acad. Sci. USA 100: Sion, S., Y. Fan and M. Tanaka, 2007 Sequential Monte Carlo without likelihood. Proc. Natl. Acad. Sci. USA 104: Tallmon, D. A., G. Luikart and M. A. Beaumont, 2004 Comparative evaluation of a new effective population ize etimator baed on approximate Bayeian computation. Genetic 167:

9 Bayeian Computation Without Likelihood 251 Tavaré, S., D. Balding, R. Griffith and P. Donnelly, 1997 Inferring coalecence time from DNA equence data. Genetic 145: Watteron, G., 1975 Number of egregating ite in genetic model without recombination. Theor. Popul. Biol. 7: Wegmann, D., C. Leuenberger and L. Excoffier, 2009 Efficient approximate Bayeian computation coupled Markov chain Monte Carlo without likelihood. Genetic 182: Wei, G., and A. von Haeeler, 1998 Inference of population hitory uing a likelihood approach. Genetic 149: Won, Y., and J. Hey, 2005 Divergence population genetic of chimpanzee. Mol. Biol. Evol. 22: Zellner, A., 1971 An Introduction to Bayeian Inference in Econometric. Wiley, New York. Communicating editor: J. Wakeley APPENDIX A: PROOFS OF THE MAIN FORMULAS To keep thi article elf-contained, we preent a proof of formula (11) and (15). The argument i an adaptation from the proof of Lemma 1 in Lindley and Smith (1972). By linearity it clearly uffice to how the formula for one fixed ampled parameter u j. The reult then follow. Theorem. Suppoe that, given the parameter vector u, the ditribution of the tatitic vector i multivariate normal, NCu 1 c 0 ; S Þ; and, given the fixed parameter vector u j, the ditribution of the parameter u i Then: 1. The ditribution of u given i u Nu j ; S u Þ: u j NTv j ; TÞ; where T ¼ C t S 1 C 1 S 1 1 u and v j ¼ C t S 1 c 0 Þ 1 S 1 u uj. 2. The marginal ditribution of i where m j ¼ c 0 1 Cu j and D ¼ S 1 CS u C t. Proof. By Baye theorem Nm j ; DÞ; pu j Þ } f j uþpuþ: The product on the right-hand ide i of the form exp 1 2 Q Þ, where Q ¼ c 0 CuÞ t S 1 ¼ u t C t S 1 C 1 S 1 u... 1 c 0 Þ t S 1 c 0 CuÞ 1 u u j Þ t S 1 u u uj Þ Þu 2 c 0Þ t S 1 c 0 Þ 1 u j Þ t S 1 u uj ¼ u t T 1 u 2v j Þ t u 1 c 0 Þ t S 1 ¼u Tv j Þ t T 1 u Tv j Þ v j Þ t Tv j u j Þ t S 1 u uj 1 c 0 Þ t S 1 c 0 Þ: Cu 1 u j Þ t S 1 u Þu 1... c 0 Þ 1 u j Þ t S 1 u uj In the lat tep we completed the quare with repect to u and ued the fact that T i ymmetric. Up to a contant that doe not depend on u j we hence get pu j Þ } cu j Þexp 1 2 u Tvj Þ t T 1 u Tv j ÞÞ ; where cu j Þ¼exp 1 2 uj Þ t S 1 u uj v j Þ t Tv j ÞÞ. Thi prove the firt part of the theorem and by linear uperpoition the validity of Equation 11.

10 252 C. Leuenberger and D. Wegmann To prove the econd part of the theorem, oberve that ¼ Cu 1 c 0 1 e with e N0; S Þ and u ¼ u j 1 h with h N0; S u Þ. Putting thee equalitie together, we get Cu j 1 c 0 1 Ch 1 e: Thi, being a linear combination of independent multivariate normal variable, i till multivariate normal with mean Cu j 1 c 0 and it covariance matrix i given by E½Ch 1 eþch 1 eþ t Š¼E½ChChÞ t 1 ee t Š¼CE½hh t ŠC t 1 E½ee t Š¼CS u C t 1 S : Thi prove the econd part of the theorem a well a formula (15). n APPENDIX B: NONLINEAR TOY MODELS In thi ection we decribe a cla of toy model that are nonlinear in the parameter u 2 R and have nonnormal, poibly heterocedatic error term. Still their likelihood are eay to calculate analytically. We et 0 1 f 1 uþ 0 1 e 1 uþ B ¼ fuþ 1 euþ ¼@.. f n uþ C B A e n uþ C A: Here f i (u) are monotonically increaing continuou function of u and e i (u) are independent, uniformly ditributed error term in the interval [ u i (u), u i (u)] 4 R, where u i (u) are nondecreaing, continuou function: e i uþ Unif½ u i uþ; u i uþšþ: It i traightforward to check that for a prior p(u) the poterior ditribution of u given ¼ 1 ;... ; n Þ t i (up to a normalizing contant) where and pu j Þ } 1 u 1 uþ... u n uþ Ind½u min; u max ŠÞpuÞ; u min ¼ max fg 1 i i i Þg; u max ¼ min i fhi 1 i Þg g i uþ ¼f i uþ 1 u i uþ; h i uþ ¼f i uþ u i uþ: For the imulation in Table 2 we choe n ¼ 5, f 1 uþ ¼¼f 5 uþ ¼u 3, and u 1 uþ¼¼u 5 uþ [ 10. The prior wa puþn0; 4Þ.