Approximate Bayesian Computational methods

Approximate Bayesian Computational methods ∗ Jean-Michel Marin † Institut de Math ´ ematiques et Mod ´ elisation (I3M), Uni versit ´ e Montpellier 2, France Pierre Pudlo Institut de Math ´ ematiques et Mod ´ elisation (I3M), Uni versit ´ e Montpellier 2, France Christian P . Robert CEREMADE, Uni versit ´ e Paris Dauphine and CREST , INSEE, Paris Robin J. Ryder CEREMADE, Uni versit ´ e Paris Dauphine and CREST , INSEE, Paris Nov ember 26, 2024 Abstract Also known as lik elihood-free methods, approximate Bayesian computational (ABC) meth- ods hav e appeared in the past ten years as the most satisfactory approach to intractable like- lihood problems, first in genetics then in a broader spectrum of applications. Howe v er , these methods suf fer to some degree from calibration difficulties that make them rather volatile in their implementation and thus render them suspicious to the users of more traditional Monte Carlo methods. In this survey , we study the v arious improv ements and extensions brought on the original ABC algorithm in recent years. Keyw ords: likelihood-free methods, Bayesian statistics, ABC methodology , DIY ABC, Bayesian model choice ∗ This research was financially supported by the French Agence Nationale de la Recherche grant ’EMILE’ ANR-09- BLAN-0145-01, as well as by the Fondation des Sciences Math ´ ematiques de P aris and a GIS scholarship for the fourth author . † Corresponding author: jean-michel.marin@uni v-montp2.fr 1 1 Intr oduction Conducting a Bayesian analysis in situations where the likelihood function ` ( θ | y ) is not av ailable raises a computational issue. The likelihood may be unav ailable for mathematical reasons (it is not a v ailable in closed from as a function of θ ) or for computational reasons (it is too expensiv e too calculate). In some specific settings, the lik elihood is expressed as an intractable multidimensional inte- gral ` ( θ | y ) = Z ` ? ( θ | y , u ) d u , where y ∈ D ⊆ R n is observed, u ∈ R p a latent vector and θ ∈ R d the parameter of interest. For instance, when facing coalecent models in population genetics (see, e.g. T av ar ´ e et al, 1997), y is the genotypes of the present sample, while u stands for their genealogical tree and the genotypes of their ancestors. In the particular set-up of hierarchical models with partly conjugate priors, it may be that the corresponding conditional distributions can be simulated and this property leads to a Gibbs sampler (Gelf and and Smith, 1990). Such a decomposition is not av ailable in general and there is no generic way to implement an MCMC algorithm like the Metropolis–Hastings algorithm (see, e.g., Robert and Casella, 2004; Marin and Robert, 2007). T ypically , the increase in dimension induced by the data augmentation from θ to u may be such that the con ver gence properties of the corresponding MCMC algorithms are too poor for the algorithm to be considered. In others situations, the normalizing constant of the likelihood Z θ is unkno wn ` ( θ | y ) = ` 1 ( θ | y ) / Z θ . This is typically the case of Gibbs random fields used to model the dependency within spatially correlated data, with applications in epidemiology and image analysis, among others (e.g. Rue and Held (2005)). For such models, a solution relying on the simulation of pseudo-samples has been proposed by Møller et al (2006). Ho we v er the dependenc y of this solution on a pseudo-tar get distribution makes it difficult to calibrate (Cucala et al, 2009; Friel and Pettitt, 2008) in general settings. Bayesian inference thus faces a lar ge class of settings where the likelihood function is not com- pletely kno wn, e.g. ` ( θ | y ) = ` 1 ( θ | y ) ` 2 ( θ ) with ` 2 unkno wn, and where exact simulation from the corresponding posterior distribution is impractical or e ven impossible. Such settings call for prac- tical if cruder approximations methods. In the past, Laplace approximations (Tierne y and Kadane, 1986) and v ariational Bayes solutions (Jaakkola and Jordan, 2000) have been advanced for such problems. Ho wev er , Laplace approximations require some analytic knowledge of the posterior distribution, while variational Bayes solutions replace the true model with another pseudo-model which is usually much simpler and thus misses some of the features of the original model. The ABC methodology , where ABC stands for appr oximate Bayesian computation , was men- tioned as early as 1984 through a pedagogical and philosophical argument in Rubin (1984). It of fers an almost automated resolution of the dif ficulty with models which are intractable but can 2 be simulated from. It w as first proposed in population genetics by T av ar ´ e et al (1997), who in- troduced approximate Bayesian computational methods as a rejection technique bypassing the computation of the likelihood function via a simulation from the corresponding distrib ution. The exact version of the method cannot be implemented but in a very small range of cases. Pritchard et al (1999) produce a generalisation based on an approximation of the target. W e study here the foundations as well as the implementation of the ABC method, with illustrations from time series. This surve y describes the genesis of the ABC approach and its justifications (Section 2), the calibration of the method (Section 3), recent sequential impro vements (Section 4), post-processing of ABC outputs (Section 5), and the specific application of ABC to model choice (Section 6). The illustrations of the ABC methodology are based on the posteriors of the MA ( 2 ) and MA ( 1 ) models for which the true posterior distribution can be computed; the impact of the ABC approximation can thus be assessed. W e do not co ver the increasingly wide array of applications of ABC here here; see Csill ` ery et al (2010a) for a survey of implementations of ABC in genomics and ecology . Neither do we address the controv ersy raised by T empleton (2008, 2010) about the lack of v alidity of the ABC approach in statistical testing. Answers to those criticisms are provided in Beaumont et al (2010); Csill ` ery et al (2010b); Berger et al (2010), among others. 2 Genesis of the ABC appr oach and justifications Prehistory Rubin (1984) advances a visionary statement that ‘Bayesian statistics and Monte Carlo methods are ideally suited to the task of passing many models over one dataset’. Further - more, he produces in this paper a description of the first ABC algorithm. Follo wed by T av ar ´ e et al (1997), the original ABC algorithm is in fact a special case of an accept-reject method (see, e.g., Robert and Casella, 2004), where the parameter θ is generated from the prior π ( θ ) and the acceptance is conditional on the corresponding simulation of a sample being ‘almost’ identical to the (true) observed sample, which is denoted y throughout this paper . For the original algorithm gi ven belo w (and solely for this algorithm), we suppose that y takes v alues in a finite or countable set D . Algorithm 1 Likelihood-free rejection sampler 1 f or i = 1 to N do repeat Generate θ 0 from the prior distribution π ( · ) Generate z from the likelihood f ( ·| θ 0 ) until z = y set θ i = θ 0 , end f or 3 It is straightforward to show that the outcome  θ 1 , θ 2 , . . . , θ N  resulting from this algorithm is an iid sample from the posterior distribution since f ( θ i ) ∝ ∑ z ∈ D π ( θ i ) f ( z | θ i ) I y ( z ) = π ( θ i ) f ( y | θ i ) ∝ π ( θ i | y ) . Rubin (1984) does not promote this simulation method in situations where the likelihood is not av ailable but rather exhibits it as an intuitiv e way to understand posterior distributions from a frequentist perspectiv e, because parameters from the posterior are more likely to be those that could hav e generated the observed data. (The issue of the zero probability of the e xact equality between simulated and observed data in continuous settings is not addressed in the original paper , presumably because the very notion of a ‘match’ between simulated and observed data is not precisely defined.) The first ABC In a population genetics setting, Pritchard et al (1999) extend the abov e algorithm to the case of continuous sample spaces, producing the first genuine ABC algorithm, defined as follo ws Algorithm 2 Likelihood-free rejection sampler 2 f or i = 1 to N do repeat Generate θ 0 from the prior distribution π ( · ) Generate z from the likelihood f ( ·| θ 0 ) until ρ { η ( z ) , η ( y ) } ≤ ε set θ i = θ 0 , end f or where the parameters of the algorithm are – η , a function on D defining a statistic which most often is not sufficient, – ρ > 0, a distance on η ( D ) , – ε > 0, a tolerance lev el. The likelihood-free algorithm abov e thus samples from the marginal in z of the joint distribu- tion π ε ( θ , z | y ) = π ( θ ) f ( z | θ ) I A ε , y ( z ) R A ε , y × θ π ( θ ) f ( z | θ ) d z d θ , (1) where I B ( · ) denotes the indicator function of the set B and A ε , y = { z ∈ D | ρ { η ( z ) , η ( y ) } ≤ ε } . 4 The basic idea behind ABC is that using a representati ve (enough) summary statistic η coupled with a small (enough) tolerance ε should produce a good (enough) approximation to the posterior distribution, namely that π ε ( θ | y ) = Z π ε ( θ , z | y ) d z ≈ π ( θ | y ) . Before moving to the extensions of the above algorithm, let us consider a simple dynamic example. Example The MA ( q ) process is a stochastic process ( y k ) k ∈ N ∗ defined by y k = u k + q ∑ i = 1 θ i u k − i , (2) where ( u k ) k ∈ Z is an iid sequence of standard Gaussians N ( 0 , 1 ) . Even though a Bayesian analysis can handle non-identifiable settings and still estimate properly identifiable quantities (see, e.g., Marin and Robert, 2007, Chapter 5), we will impose a standard identifiability condition on this model, namely that the roots of the polynomial Q ( x ) = 1 − q ∑ i = 1 θ i x i are all outside the unit circle in the comple x plane. A simple prior distrib ution is therefore the uniform distrib ution o ver the corresponding range of θ i ’ s, especially when q is small and the set of resulting parameters is easy to describe. In the case processed in the figures below for q = 2, we obtain the triangle − 2 < θ 1 < 2 , θ 1 + θ 2 > − 1 , θ 1 − θ 2 < 1 . Although the prior on θ is very simple, and despite the Gaussian nature of the random vari- ables, the likelihood associated with a series ( y k ) 1 ≤ k ≤ n is more complex because of the need to integrate out u − q + 1 , . . . , u − 1 , u 0 . (The easier alternativ e is to condition on ( y k ) 1 ≤ k ≤ q , see Marin and Robert, 2007, e ven though the general case can also be handled by MCMC simulations as the likelihood is a v ailable, at least for small v alues of n .) Running one iteration of ABC in this setting then simply requires (a) simulating the MA ( q ) coefficients θ uniformly ov er the acceptable range, (b) generating an iid sequence ( u k ) − q < k ≤ n , (c) producing a simulated series ( z k ) 1 ≤ k ≤ n . 5 ● −2 −1 0 1 2 −1.0 −0.5 0.0 0.5 1.0 θ 1 θ 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Figure 1: Comparison of the le vel sets (in black) of the true posterior distrib ution with the scatter plot (in blue) of an ABC sample when using autocovariances as summary statistics. The threshold ε is chosen so that 0 . 1% of the N = 10 6 simulated datasets are accepted. The observed dataset has been drawn from an MA ( 2 ) model with n = 100 epochs and parameter θ = ( 0 . 6 , 0 . 2 ) (the red dot) . The triangle is the range of acceptable values of θ . 6 Depending on the focus of the analysis, the distance can be the raw distance between the series ρ 2 { ( z k ) 1 ≤ k ≤ n , ( y k ) 1 ≤ k ≤ n } = n ∑ k = 1 ( y k − z k ) 2 or the quadratic distance between summary statistics like the first q autocov ariances τ j = n ∑ k = j + 1 y k y k − j which is our choice for the illustration provided in Figure 1. This e xperiment sho ws ho w an ABC sample fits the lev el sets of the true posterior density for a simulated sample of length 100 using the parameters ( θ 1 , θ 2 ) = ( 0 . 6 , 0 . 2 ) and a tolerance lev el equal to the 0 . 1% quantile of the sample of the distances. (The level sets were computed from the exact likelihood for the MA ( 2 ) model and a grid of v alues of θ over the acceptable range.) This plot illustrates how the distribution of the sample points departs from true posterior: the approximation does not reconstruct the posterior perfectly . Decreasing ε would lead to a better concentration of the posterior density on the lev el sets, but at the e xpense of the size of the resulting sample or at a higher computing cost. J MCMC-ABC In practice, using simulations from the prior distribution π ( · ) is inef ficient be- cause this does not account for the data at the proposal stage and thus leads to proposed values located in low posterior probability regions. As an answer to this problem, Marjoram et al (2003) introduce an MCMC-ABC algorithm (Algorithm 3) tar geting the approximate posterior distribu- tion π ε of equation (1). Algorithm 3 Likelihood-free MCMC sampler Use Algorithm 2 to get a realisation ( θ ( 0 ) , z ( 0 ) ) from the ABC target distribution π ε ( θ , z | y ) f or t = 1 to N do Generate θ 0 from the Marko v kernel q  ·| θ ( t − 1 )  , Generate z 0 from the likelihood f ( ·| θ 0 ) , Generate u from U [ 0 , 1 ] , if u ≤ π ( θ 0 ) q ( θ ( t − 1 ) | θ 0 ) π ( θ ( t − 1 ) ) q ( θ 0 | θ ( t − 1 ) ) and ρ { η ( z 0 ) , η ( y ) } ≤ ε then set ( θ ( t ) , z ( t ) ) = ( θ 0 , z 0 ) else ( θ ( t ) , z ( t ) ) = ( θ ( t − 1 ) , z ( t − 1 ) ) , end if end f or 7 The acceptance probability used in Algorithm 3 does not inv olve the calculation of the likeli- hood and it thus satisfies ABC requirements. It also produces an MCMC algorithm which e xactly targets π ε ( θ , z | y ) as its stationary distribution. Indeed, π ε ( θ 0 , z 0 | y ) π ε ( θ ( t − 1 ) , z ( t − 1 ) | y ) × q ( θ ( t − 1 ) | θ 0 ) f ( z ( t − 1 ) | θ ( t − 1 ) ) q ( θ 0 | θ ( t − 1 ) ) f ( z 0 | θ 0 ) = π ( θ 0 ) f ( z 0 | θ 0 ) I A ε , y ( z 0 ) π ( θ ( t − 1 ) ) f ( z ( t − 1 ) | θ ( t − 1 ) ) I A ε , y ( z ( t − 1 ) ) × q ( θ ( t − 1 ) | θ 0 ) f ( z ( t − 1 ) | θ ( t − 1 ) ) q ( θ 0 | θ ( t − 1 ) ) f ( z 0 | θ 0 ) = π ( θ 0 ) q ( θ ( t − 1 ) | θ 0 ) π ( θ ( t − 1 ) q ( θ 0 | θ ( t − 1 ) ) I A ε , y ( z 0 ) . The initialisation of the MCMC sampler with the rejection sampler (Algorithm 2) can be by- passed since the Markov chain for gets its initial state. The computational cost of the initialisation is then reduced. But then we hav e to run the MCMC longer to achiev e con ver gence and omit the burn-in first iterations from the output, which also has a computational cost. As noted above, the ABC approximation depends on tuning parameters (the summary statistic η , the tolerance ε , and the distance ρ ) that ha ve to be chosen prior to running the algorithm and the calibration of which is discussed in most of the literature. The tolerance ε is some what the easiest aspect of this calibration issue in that, when ε goes to zero, the ABC algorithm becomes e xact. Noisy ABC W ilkinson (2008) proposes to switch perspectiv e, replacing the approximation error resulting from the loose acceptance condition in the above likelihood-free samplers with an exact inference from a controlled approximation of the target, essentially a con volution of the re gular target with an arbitrary k ernel function. The corresponding ABC target is thus π ε ( θ , z | y ) = π ( θ ) f ( z | θ ) K ε ( y − z ) R π ( θ ) f ( z | θ ) K ε ( y − z ) d z d θ , (3) where K ε is a well-chosen kernel parameterised by the bandwidth ε . This perspecti ve is interesting in that the outcome is completely controlled, due to the de gree of freedom brought by the choice of the kernel. W ilkinson (2008) makes the v aluable point that if the model includes an error term, then taking the distribution of that error term to be K ε leads to an ABC algorithm which simulates exactly from the error-in-v ariables posterior . In practice, W ilkinson’ s (2008) approach requires a modification of the standard ABC algorithms, taking into account the kernel K ε for the simulation of z . The new algorithm which includes an accept-reject step imposes an upper bound on the con volution kernel K ε . This perspectiv e of the “noisy ABC” is also adopted by Fearnhead and Prangle (2010) who study the con vergence of ABC based inference. They show that the con volution induced by 8 the kernel representation leads to the true parameter being the maximum of the integrated log- likelihood and thus that a Bayes estimator is con verging to the true v alue when the number of observ ations goes to infinity and the tolerance level goes to zero. They also stress the connection with the econometrics approach of indirect inference Gouri ´ eroux et al (1993). ABC Filtering Jasra et al (2011) propose an ABC scheme for filtering when the distribution of the observ ables conditioned on the hidden state is not av ailable point-wise, related to the con v olu- tion particle filter of Campillo and Rossi (2009). It is particularly appealing in that it allows com- plex (hence realistic) statistical models for filtering. Theoretical arguments are given to prov e that the ABC approximation of the filter does not accumulate errors along the sequence of observables, when the model has good mixing properties. Dean et al (2011) illustrate this implementation in the specific case of hidden Markov (HMM) models, relating the ABC implementation with W ilkin- son’ s (2008) perspecti ve and demonstrating that the pseudo (or noisy) model for which ABC is exact also is an HMM. Using this representation, they further establish ABC consistency . While Dean et al (2011) establish that ABC leads to an asymptotic bias for a fixed value of the toler - ance ε , they also prov e that an arbitrary accuracy can be attained with enough data and a small enough ε . (W e note that the restriction to summary statistics that preserv e the HMM structure is paramount for the results in the paper to apply , hence pre venting the use of truly summarising statistics that would not gro w in dimension with the size of the HMM series.) The con ver gence result central to Dean et al (2011) is also connected with Fearnhead and Prangle’ s (2010) version, mentioned abov e, in that they both rely on pseudo-lik elihood consistency ar guments. 3 Calibration of ABC Summary statistics Sev eral authors ha ve considered the fundamental difficulty associated with the choice of the summary statistic, η ( y ) , which one would like to consider as a quasi-sufficient statistic. First, for most real problems (a notable e xception being found in Grelaud et al, 2009 in the case of Gibbs random fields), it is impossible to find non-trivial suf ficient statistics which would eliminate the need of a choice of statistics. Second, the summary statistics of interest are usually determined by the problem at hand and chosen by the experimenters in the field. Assuming a lar ge collection of summary statistics is av ailable, Joyce and Marjoram (2008) consider the sequential inclusion of those statistics into the ABC target. The inclusion of a ne w statistic within the set of summary statistics is assessed in terms of a likelihood ratio test, without taking into account the sequential nature of the tests. W e hav e reservations about the method, first and foremost that the construction of the statistics is not discussed, while the method is not independent from parametrisation, and also that the order in which the statistics are considered is paramount for their inclusion/exclusion. A regularisation of the method proposed at the end of the paper is to use a forward-backward selection mechanism to address this last issue. Ho wev er, this correction does not address another issue, namely the impact of the correlation between the summary statistics. Note at last that Joyce and Marjoram’ s (2008) method still depends on an 9 approximation factor that needs to be calibrated prior to running the algorithm. In his thesis, Ratmann (2009) proposes a similar examination of the successi ve inclusion of various statistics. A related perspecti ve is that of McKinley et al (2009). They perform a simulation experi- ment comparing ABC-MCMC and ABC-SMC (discussed below) with regular data augmentation MCMC. The authors test strategies to select the tolerance le vel, and to choose the distance ρ and the summary statistics. The conclusions are not very surprising, in that (a) repeating simulations of the data points giv en one simulated parameter does not seem to contribute to an impro ved approximation of the posterior by the ABC sample, (b) the tolerance level does not seem to ha ve a strong influence, (c) the choice of the distance, of the summary statistics and of the calibration factors are paramount to the success of the approximation, and (d) ABC-SMC outperforms ABC-MCMC (MCMC remaining the reference). Fearnhead and Prangle (2010) study the selection of summary statistics with the interesting perspecti ve that ABC is then considered from a purely inferential viewpoint and calibrated for estimation purposes. (This contrasts with most alternati ve perspectiv es that en vision ABC as a poor man’ s non-parametric estimation of the posterior distribution.) Fearnhead and Prangle (2010) rely on a randomised version of the summary statistics from which they deriv e a well- calibrated version of ABC, i.e. an algorithm that gi ves proper predictions of gi ven quantities. The authors consider choices of summary statistics, and establish that the posterior expectations of the parameters of interest are optimal summary statistics, although this follows from their choice of loss function. T olerance threshold and ABC appr oximation error As noted abo ve, the choice of the toler - ance le vel ε is mostly a matter of computational po wer: smaller ε ’ s are associated with higher computational costs and the standard practice (Beaumont et al, 2002) is to select ε as a small percentile of the simulated distances ρ { η ( z ) , η ( y ) } . An alternati ve described below is to set the ABC algorithm within the non-parametric setting of density estimation, in which case ε is understood as a bandwidth and can be deriv ed from the simulated population. As noted in Fearnhead and Prangle (2010), this perspective implies that the optimal ε is then dif ferent from zero. Standing rather apart from other contributions to the field, Ratmann et al (2009) provide an intrinsically nov el way of looking at the ABC approximation error (and hence at the tolerance). It is presented as a tool assessing the goodness of fit of a gi ven model. The fundamental idea there 10 is to use the tolerance ε as an additional parameter of the model, simulating from a joint posterior distribution f ( θ , ε | y ) ∝ ξ ( ε | y , θ ) π θ ( θ ) π ε ( ε ) , where ξ ( ε | y , θ ) plays the role of the lik elihood, and π θ and π ε are the corresponding priors on θ and ε . In this approach, ξ ( ε | y , θ ) is the prior predictiv e density of ρ { η ( z ) , η ( y ) } gi ven θ and y when z is distributed from f ( z | θ ) . W e note here a connection with W ilkinson’ s (2008) target (3) in that π ( θ ) f ( z | θ ) K ε ( y − z ) is identical to the abov e once we replace y − z by ε . Ratmann et al (2009) then deri ve an ABC algorithm they call ABC µ to simulate an MCMC chain targeting this joint distrib ution, replacing ξ ( ε | y , θ ) with a non-parametric kernel approxi- mation. For each model under comparison, the marginal posterior distribution on the error ε is then used to assess the fit of the model, the logic being that this posterior should include 0 in a reasonable credible interv al. While the authors stress they use the data once, they also define the abov e tar get by using simultaneously a prior distribution on ε and a conditional distrib ution on the same ε that they interpret as the lik elihood in ( ε , θ ) . The product is most often defined as a density in ( ε , θ ) , so it can be simulated from, b ut the Bayesian interpretation of the outcome is delicate, especially because it seems the prior on ε contributes significantly to the final assessment of the model. As discussed in Robert et al (2010), some of the choices of Ratmann et al (2009) can be argued about, in particular the ambiv alent role of the approximation error . The most important aspect of the paper is that the original motiv ation of running ABC for conducting inference on the parameters of a model is replaced by the alternati ve goal of running ABC for assessing a model; see Ratmann et al’ s 2010 reply to the remarks made by Robert et al (2010). . Example Returning to the MA ( 2 ) model, we study the impact of the choice of the distance and of the tolerance on the approximation. In this example, we simulated a sample of size 50 from a MA ( 2 ) model based on the same parameters as above. First, we compare the impact of using the raw distance between the complete datasets instead of the distance between the autocovariances (introduced abov e). Figure 2 shows that the raw distance between the observed and the simulated time series is inefficient and fairly non-discriminativ e. F or the raw distance, the spread of the parameters accepted after the ABC step is indeed much wider than for the second distance, espe- cially when compared with the level sets of the posterior density . W e thus use only the distance between the autocov ariances in the remainder of the paper . W e no w turn to the tolerance ε . Figure 3 sho ws that decreasing ε along empirical quantiles of the simulated distances ρ ( η ( z ) , η ( y )) impro ves the approximation, although we nev er reach the true marginal densities (this is particularly true for the parameter θ 2 .) The marginal densities of the ABC samples were obtained by the R default density estimator and the true marginal densities by numerical integration. J 11 ● −2 −1 0 1 2 −1.0 −0.5 0.0 0.5 1.0 θ 1 θ 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 −1 0 1 2 −1.0 −0.5 0.0 0.5 1.0 θ 1 θ 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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The lev el sets of the posterior density are exhibited in black . 12 −2 −1 0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 θ 1 −1.0 −0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 θ 2 Figure 3: Evolution of the distribution of ABC samples using different quantiles for ε (10% in blue , 1% in r ed , and 0 . 1% in yellow ) when compared with the true mar ginal densities. The dataset is the same as in Figure 2. 4 Sequential impr ovements Importance sampling Sequential techniques can enhance the efficiency of the ABC algorithm by learning about the tar get distribution, as in Sisson et al’ s (2007) partial rejection control (PRC) version. The ABC-PRC modification introduced by Sisson et al (2007) consists in producing sam- ples ( θ ( t ) 1 , . . . , θ ( t ) N ) at each iteration 1 ≤ t ≤ T of the algorithm by using a particle filter method- ology . Starting with a regular ABC step, the generation of the θ ( t ) i ’ s relies on Mark ov transition kernels K t , θ ( t ) i ∼ K t ( θ | θ ? ) , until z ∼ f ( z | θ ( t ) i ) is such that ρ ( η ( z ) , η ( y )) ≤ ε , where θ ? is selected at random among the pre- vious θ ( t − 1 ) i ’ s with probabilities ω ( t − 1 ) i . The probability ω ( t ) i is deriv ed by an importance sampling argument as ω ( t ) i ∝ π ( θ ( t ) i ) L t − 1 ( θ ? | θ ( t ) i ) π ( θ ? ) K t ( θ ( t ) i | θ ? ) , where L t − 1 is an arbitrary transition kernel. While this method is based upon the theoretical work of Del Moral et al (2006) and their SMC sampler, the application to approximate Bayesian compu- tation results in a bias in the approximation to the posterior , because the lik elihood is removed in a standard ABC fashion (Sisson et al, 2009). Replacing the likelihood with the indicator function 13 provides an unbiased estimator of the likelihood that cannot be used as such in the denominator of a Metropolis–Hastings acceptance probability , hence the resulting bias. An alternative version called ABC-PMC and based on genuine importance sampling argu- ments, proposed by Beaumont et al (2009), bypasses this dif ficulty , in connection with the popu- lation Monte Carlo method of Douc et al (2007). It includes an automatic scaling of the forward kernel. The correction published in Sisson et al (2009) acknowledges the e xistence of a bias and suggests a correction essentially identical to the PMC solution of Beaumont et al (2009). As illustrated in the pseudo-code belo w , ABC-PMC constructs a kernel approximation to the target distribution based on earlier simulations and estimates the random w alk scale (which is also the kernel bandwidth) from those simulations, using in addition a decreasing sequence of tolerance thresholds ε 1 ≥ . . . ≥ ε T : Algorithm 4 Likelihood-free population Monte Carlo sampler At iteration t = 1, f or i = 1 to N do repeat Simulate θ ( 1 ) i ∼ π ( θ ) and z ∼ f ( z | θ ( 1 ) i ) until ρ ( η ( z ) , η ( y )) ≤ ε 1 Set ω ( 1 ) i = 1 / N end f or T ake Σ 1 as twice the empirical v ariance of the θ ( 1 ) i ’ s f or t = 2 to T do f or i = 1 to N do repeat Pick θ ? i from the θ ( t − 1 ) j ’ s with probabilities ω ( t − 1 ) j Generate θ ( t ) i ∼ N ( θ ? i , Σ t − 1 ) and z ∼ f ( z | θ ( t ) i ) until ρ ( η ( z ) , η ( y )) ≤ ε t Set ω ( t ) i ∝ π ( θ ( t ) i ) / ∑ N j = 1 ω ( t − 1 ) j ϕ n Σ − 1 / 2 t − 1  θ ( t ) i − θ ( t − 1 ) j o end f or T ake Σ t as twice the weighted v ariance of the θ ( t ) i ’ s end f or Another related paper is T oni et al’ s 2009 proposal of a parallel sequential ABC algorithm. Just like ABC-PMC, the ABC-SMC algorithm (an acron ym found in sev eral papers) de veloped therein is based on a sequence of simulated samples, Mark ov transition kernels, and importance weights rather than SMC justifications. The unav ailable lik elihood is estimated by the indicator of the tolerance zone or an av erage of indicators as in Marjoram et al (2003). The bulk of the paper is dedicated to the analysis of ODEs, using uniform distributions as transition kernels. The adapti vity of the ABC-SMC algorithm is restricted to a progressiv e reduction of the tolerance, ε t , since the 14 kernels K t ’ s remain the same across iterations, in contrast with the ABC-PMC motiv ation for tuning the K t ’ s to the target. The paper also contains a comparison with ABC-PRC, which shows a bias in the v ariance of the ABC-PRC output, in line with Beaumont et al (2009). McKinley et al (2009) hav e coded the parallel sequential ABC algorithm on an infectious dis- ease model (a recent outbreak of Ebola Haemorrhagic Fe ver in the Democratic Republic of the Congo — for which there is no known treatment and which is responsible for an 88% decline in observed chimpanzee populations since 2003!). They show that the ABC-SMC sampler out- performs ABC-MCMC (MCMC remaining the reference). The comparison e xperiment is based on a single dataset, with fixed random walk v ariances for the MCMC algorithms; note that the prior used in the simulation might be too highly peaked around the true v alue (gamma rates of 0 . 1). Some of the ABC scenarios do produce estimates that are rather far away from the refer- ences gi ven by MCMC, for instance CABC-MCMC when the threshold ε is 10 and the number of repeats R is 100. Backward kernels and SMC Del Moral et al (2009) exhibit the connection between the ABC algorithm and the foundational SMC paper of Del Moral et al (2006) that inspired Sisson et al (2007). As opposed to the latter , and despite a common frame work, this ABC-SMC paper properly relies on the idea of using backward kernels L t to simplify the importance weights and to remove from these weights the dependence on the unknown lik elihood. A major assumption of Del Moral et al (2009) is that the forward kernels K t are supposed to be in variant against the true tar get (which is a tempered-like version of the true posterior in sequential Monte Carlo), a choice not explicitely made in Sisson et al (2007). One of the nov elties in the paper is that the authors rely on M repeated simulations of the pseudo-data z giv en the parameter , rather than using a single simulation. In that perspectiv e, each simulated parameter gets a non-zero weight that is proportional to the number of accepted z ’ s. The limiting case M → ∞ brings in an e xact simulation from the tempered targets π ε t ’ s, so there is a con ver gence principle and the stabilisation of the approximation could be assessed to calibrate M . The adaptivity in the ABC-SMC algorithm is found in the on-line construction of the thresholds: the thresholds decrease slo wly enough to keep a large number of accepted transitions from the pre vious sample. An important feature is that the update in the importance weights simplifies to the ratio of the proportions of survi ving particles, due to the choice of the re versal backward kernels L t and to the use of in variant transition forward kernels K t . In a very related manner , Drov andi and Pettitt (2010) use a combination of particles and of MCMC mov es to adapt a proposal to the true target, with acceptance probability min  1 , π ( θ ∗ ) K ( θ c | θ ∗ ) π ( θ ∗ ) K ( θ ∗ | θ c )  where θ ∗ is the proposed value, θ c is the current value (picked at random from the particle popu- lation), and K is a proposal kernel used to simulate the proposed value. The algorithm is adaptive in that the pre vious population of particles is used to mak e the choice of the proposal K , as well as 15 of the tolerance lev el ε t . The le vel of novelty of the method compared with Del Moral et al (2009) is quite limited, since the paper adapts the tolerance on-line as an α -quantile of the previous par - ticle population. The conv er gence analysis which is omitted by Drovandi and Pettitt (2010) is perhaps not so standard, mainly because the MCMC is applied only to half of the particle system. Del Moral et al (2011) tackle the issue of adaptiv e resampling strategies. The only strong method- ological difference between the two papers is that the MCMC steps are now repeated ‘numerous times’. Howe ver , this partly cancels the appeal of an O ( N ) order method versus the O ( N 2 ) order ABC-PMC and ABC-SMC methods. An interesting remark there is that advances are needed in cases when simulating the pseudo-observations is very costly , as in Ising models. Ho we ver , re- placing e xact simulation by a fe w steps from a Gibbs sampler as in Grelaud et al (2009) cannot be very detrimental to the con ver gence of an approximate algorithm. 5 P ost-processing of ABC output Local linear regr ession Improv ements to the general ABC scheme ha ve been achiev ed by view- ing the problem as a conditional density estimation and dev eloping techniques to allow for larger ε (Beaumont et al, 2002). This is a post-processing scheme in that the simulation process per se does not change b ut the analysis of the ABC output does. The authors endea vour to include all simulated summary statistics, ev en those f ar away from the observ ed summary statistic, by shrink- ing the corresponding parameters in a linear manner . More specifically , they replace the simulated θ ’ s with θ ∗ = θ − { η ( z ) − η ( y ) } T ˆ β , where ˆ β is obtained by a weighted least squares regression of θ on ( η ( z ) − η ( y )) , using weights of the form K δ { ρ { η ( z ) , η ( y ) } } , where K δ is a non-parametric kernel with bandwidth δ . Example W e implement this correction of Beaumont et al (2002) in the MA ( 2 ) model, again using the first tw o autocov ariances as summary statistic η ( z ) , and we apply a non-parametric local regression based on the Epanechnik ov kernel. W e keep δ equal to the v alue of the tolerance ε used in the regular ABC scheme. Figures 4 and 5 summarise the results. When using a 0 . 1% quantile, the tw o density estimates are identical in the case of the parameter θ 2 . The post-processed density estimate of θ 1 is closer to the true posterior . When using a 20% quantile, the impact of the local regression is more spectacular . W e recov er results close to those obtained with the 0 . 1% quantile. This exhibits the point that local regression strongly attenuates the impact of the truncation brought by ε . J 16 −2 −1 0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 θ 1 −2 −1 0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 θ 2 Figure 4: Comparison of the density estimates of the distributions of the parameters using an ABC approximation with ε as the 0 . 1% quantile on the autocov ariance distances (in blue) and the Beaumont et al (2002) correction (in r ed) . The red and blue curv es are confounded for the parameter θ 2 . 17 −2 −1 0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 θ 1 −2 −1 0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 θ 2 Figure 5: Comparison of the approximate distributions of the parameters using an ABC approxi- mation with ε as the 20% quantile on the autocov ariance distance (in blue) and the Beaumont et al (2002) correction (in red) . Nonlinear r egression Blum and Franc ¸ ois (2010) propose a generalisation of Beaumont et al’ s (2002) ABC post-processing where the local linear regression of the parameter θ on the summary statistics η ( z ) is replaced by a nonlinear re gression with heteroskedasticity . In this ne w approach, the nonlinear mean and variance are estimated by a neural net with one hidden layer, using the R package nnet (R Dev elopment Core T eam, 2006). The result is interesting in that it seems to allo w for the inclusion of more or ev en all the simulated pairs ( θ , z ) , compared with Beaumont et al. (2002). This is someho w to be e xpected since the nonlinear fit adapts dif ferently to different parts of the space. Therefore, weighting simulated ( θ , z ) ’ s by a k ernel K δ ( z − y ) is not very relev ant and it is thus not surprising that the bandwith δ is not influential, in contrast with basic ABC and ev en Beaumont et al. (2002) where δ has a different meaning. The non-parametric perspective adopted in the paper is nonetheless of the highest importance, as it prov es the most fruitful approach to the interpretation of ABC methods. In connection with this paper , Blum (2010) provides a good revie w of the non-parametric handling of ABC techniques. The true dif ficulty with the non- parametric perspecti ve lies with the curse of dimensionality . This issue might be addressed by mixing dimension reduction with recycling by shrinking as in Beaumont et al. (2002). In verse regr ession Leuenberger et al (2010) also relate to the local regression ideas in Beaumont et al. (2002). As in the earlier work by W ilkinson (2008), the approximation to the distrib ution of the parameters gi ven the observ ed summary statistics is central to the paper . In opposition to 18 Beaumont et al (2002), there is no clear shrinkage for summary statistics that are far away from the observed summary statistics: all accepted parameters are weighted similarly in the Gaussian linear approximation to the truncated prior . The other dif ference with Beaumont et al (2002) is that the authors model z giv en θ rather than θ gi ven z , in an in verse regression perspecti ve, followed by a sort of Laplace approximation reminding Rue et al (2009). 6 ABC and model choice 6.1 Bayesian model choice Model choice is one particular aspect of Bayesian analysis that inv olves computational comple x- ity , if only because several models are considered simultaneously (see, e.g., Robert, 2001; Marin and Robert, 2010). In addition to the parameters of each model, the inference considers the model index M , which is associated with its o wn prior distrib ution π ( M = m ) ( m = 1 , . . . , M ) as well as a prior distribution on the parameters conditional on the v alue m of the model index, π m ( θ m ) , defined on the parameter space Θ m . The choice between these models is then driven by the pos- terior distribution of M , a challenging computational target where ABC brings a straightforward solution. Indeed, once M is incorporated within the parameters, the ABC approximation to the posterior follo ws from the same principles as regular ABC, as sho wn by the following pseudo- code, where η ( z ) = ( η 1 ( z ) , . . . , η M ( z )) is the concatenation of the summary statistics used for all models (with elimination of duplicates). Algorithm 5 Likelihood-free model choice sampler (ABC-MC) f or i = 1 to N do repeat Generate m from the prior π ( M = m ) Generate θ m from the prior π m ( θ m ) Generate z from the model f m ( z | θ m ) until ρ { η ( z ) , η ( y ) } < ε Set m ( i ) = m and θ ( i ) = θ m end f or The ABC estimate of the posterior probability π ( M = m | y ) is then the acceptance frequency from model m , namely 1 N N ∑ i = 1 I m ( i ) = m . This also corresponds to the proportion of simulated datasets that are closer to the data y than the tolerance ε . Cornuet et al (2008) follo w the rationale that led to the local linear regression in Beaumont et al (2002) and rely on a weighted polychotomous logistic regression to estimate 19 π ( M = m | y ) . This modeling clearly brings some further stability to the abo ve estimate of π ( M = m | y ) and is implemented in the DIY ABC software described in Cornuet et al (2008) . Example Returning once again to our benchmark MA ( 2 ) model, we compare the computation of the model posterior probabilities based on an ABC sample (acceptance frequency within each model) with the true value of the Bayes f actor , which was obtained by numerical integration. The dataset used in the experiment is a time-series simulated and we wish to choose between two models: an MA ( 2 ) or an MA ( 1 ) model. Figure 6 sho ws our estimates for data simulated from ar MA ( 2 ) model. The weight of the MA ( 2 ) model increases slightly as ε decreases. Howe ver , e ven for the quantile at 0 . 01% the estimated posterior probability for the MA(2) model is equal to 0 . 72 which is far from the true v alue 0 . 95. Figure 7 shows a similar phenomenon for data simulated from an MA ( 1 ) model. J 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 Figure 6: Boxplots of the ev olution [against ε ] of ABC approximations to the Bayes fac- tor . The representation is made in terms of frequencies of visits to [accepted proposals from] models MA ( 1 ) (left) and MA ( 2 ) (right) during an ABC simulation when ε corresponds to the 10 , 1 , 0 . 1 , 0 . 01% quantiles on the simulated autocov ariance distances. The data are the same as in Figure 5. The true Bayes factor B 21 is equal to 17 . 71, corresponding to posterior probabilities of 0 . 05 and 0 . 95 for the MA(1) and MA(2) models respecti vely . The discrepancy in the above e xample shows the limitations of the ABC approximation of Bayes factors exposed in Robert et al (2011). While we could expect to obtain a better approxi- mation with a massive computational ef fort, it may be that the use of different summary statistics for dif ferent models pre vents us from con verging to the true v alue. In other words, the concatena- 20 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 1 2 0.0 0.2 0.4 0.6 0.8 1.0 Figure 7: Boxplots of evolution of Bayes factor approximations in terms of frequencies of visits to models MA ( 1 ) (left) and MA ( 2 ) (right) using an ABC approximation with 10 , 1 , . 1 , . 01% quantiles on the autocovariance distance as ε . The dataset is a sample of 50 points from a MA ( 1 ) model with θ 1 = 0 . 6. The true Bayes factor B 21 is equal to . 004 corresponding to posterior probabilities of 0 . 996 and 0 . 004 for the MA(1) and MA(2) models respecti vely . tion of sufficient statistics for individual models does not al ways constitute a suf ficient statistic for model choice, as discussed in the next paragraph. 6.2 The case of Gibbs random fields Grelaud et al (2009) show that, for Gibbs random fields and in particular for Potts models, where the goal is to compare se veral neighbourhood structures, the computation of the posterior proba- bilities of the models under competition can be operated by lik elihood-free simulation techniques. W e recall first that Gibbs random fields are probabilistic models associated with the likelihood function ` ( θ | y ) = 1 Z θ exp { θ T η ( y ) } , where y is a vector of dimension n taking values ov er a finite set X (possibly a lattice), η ( · ) is the potential function defining the random field, taking v alues in R p , θ ∈ R p is the associated parameter , and Z θ is the corresponding normalising constant. A special b ut important case of Gibbs random fields is associated with a neighbourhood structure denoted by i ∼ i 0 (meaning that i and i 0 are neighbours), in that η ( y ) = ∑ i 0 ∼ i I { y i = y i 0 } , 21 where ∑ i 0 ∼ i indicates that the summation is over all the pairs of neighbours. In that case, θ is a scalar . The central property ensuring an ABC resolution for Gibbs random fields is that, due to their exponential f amily structure, there exists a suf ficient statistic vector that runs across models and which allows for an exact ( ε = 0) simulation from the posterior probabilities of the models. Indeed, model choice in volves M Gibbs random fields in competition; each field is associated with a potential function η m ( 1 ≤ m ≤ M ) , i.e . with the corresponding likelihood ` m ( θ m | y ) = exp  θ T m η m ( y )   Z θ m , m , where θ m ∈ Θ m and Z θ m , m is the unkno wn normalising constant. From a Bayesian perspecti ve, considering an extended parameter space Θ = ∪ M m = 1 { m } × Θ m that includes the model index M , the computational target is thus the model posterior probability π ( M = m | y ) ∝ Z Θ m ` m ( θ m | y ) π m ( θ m ) d θ m π ( M = m ) , i.e. the marginal in M of the posterior distribution on ( M , θ 1 , . . . , θ M ) giv en y . Each model has its own sufficient statistic η m ( · ) . Then, for each individual model, the vector of statistics η ( · ) = ( η 1 ( · ) , . . . , η M ( · )) is clearly sufficient. Howe ver Grelaud et al (2009) exposed the fact that η is also suf ficient for the joint parameter ( M , θ 1 , . . . , θ M ) . That the concatenation of the sufficient statistics of each model is also a sufficient statistic for the joint parameter across models is clearly a property that is specific to e xponential families. As sho wn by Didelot et al (2011), ABC-based model choice can process exponential families by creating inter -model suf ficient statistics that incorporate the intra-model sufficient statistics as well as possibly the dominating measures for all models. The Gibbs random field abo ve is a specific case of this sufficienc y . Ho wever , outside exponential families, the possibility of creating a sufficient statistic of a dimension that is much lower than the dimension of the data is impossible, as explained in Robert et al (2011). 6.3 General issues T oni et al (2009) and T oni and Stumpf (2010) re view ABC-based model choice, inclusi ve of the abov e Gibbs random field example. The authors study in particular the consequences of imple- menting a sequential algorithm like ABC-PMC in this set-up. The ABC algorithm is modified to incorporate the model index, resorting to the previous assessment of π ( M = m | y ) to propose the model indices of the next population. The importance sampling features of this setting imply that the posterior probability can be estimated from the importance weights. Howe ver , the adapti vity at the core of ABC-PMC and ABC-SMC implies adapting an approximation kernel for each model. As most other perspecti ves on ABC, T oni and Stumpf (2010) do not question the role of the ABC distance in model choice settings. The Bayes factors are observ ed to be sensitiv e to the choice of the prior distributions, of the tolerance le vels, and to the variances of the kernels K t (see Section 22 4), a dependence that should not occur , since this is a simulation parameter that is unrelated with the statistical problem. It is worth pointing out the remark made by Leuenberger et al (2010) about model choice and the use of the approximation of the normalising constant resulting from the modelling to get to the marginal likelihood and the computation of the Bayes factor . This relates to earlier comments in the literature about the ABC acceptance rate approximating the marginal and a recent paper by Bartolucci et al (2006) studying w ays of computing marginal probabilities by Rao–Blackwellising re versible jump acceptance probabilities. Grelaud et al (2009) also mak e the most of this ABC feature for Ising models, since an e xact ABC (corresponding to ε = 0) algorithm is then a v ailable for model selection. A (minor) Bayesian issue mentioned by Ratmann et al (2009) is the f act that both θ and ε are taken to be the same across models. In a classical Bayesian perspective, modulo the reparam- eterisation, θ cannot be entirely dif ferent from one model to the ne xt, but using the same prior on ε ov er all models under comparison is more of an issue. The paper also considers the impact of testing for the adequac y of a model as testing for the hypothesis H 0 : ε = 0, an interesting if controv ersial stance, since e ven when the model fits, ε necessarily varies around zero. At this stage, the most perplexing feature of ABC model choice is the lack of con ver gence guarantees. As exposed in Robert et al (2011), most settings where ABC model choice is imple- mented do not allow for inter -model sufficiency in the selection of the summary statistics, because some models are not within exponential families and because using the whole data is too demand- ing. As shown by the MA e xample abov e. this lack of sufficienc y may be quite detrimental to the quality of the ABC approximation of the Bayes factors. There is therefore currently no theoretical support for the use of ABC approximations of Bayes factors and posterior model probabilities, and we thus advise for more empirical assessments in the spirit of Ratmann et al (2009) that ev aluate the model fit within each model without concluding by exact figures of the probabilities of the dif ferent models. 7 Discussion Approximate Bayesian Computation allows inference from a wide class of models which would otherwise be una vailable. As such, it has spa wned interest in both theoretical issues and applica- tions. Recent adv ances re garding the calibration of the method lead to an approximation that is good enough to be highly useful in many situations. The ef ficiency of the method can be greatly improv ed with sequential techniques and post-processing regression on the output. Nonetheless, ABC is not a silver bullet. In the current state of the art, it can only be used for model choice in a limited range of models. Future advances must at the same time expand further the tools to make ABC useful in a wider class of models, extend pre- and post-processing methods to control the approximation, and establish more clearly in which cases ABC reaches its limitations. 23 ABC methods are currently under an intense scrutiny by both statisticians and practitioners, hence the object of an unparalleled dev elopment. While this rapid de velopment pro vides answers to some interrogations from the statistical community about the v alidity of the approach and from the practitioners about a higher efficienc y of the method, some issues remain unsolv ed, among which: • the conv ergence results obtained so far are unpractical in that they require either the tol- erance to go to zero or the sample size to go to infinity . Obtaining exact error bounds for positi ve tolerances and finite sample sizes would bring a strong improv ement in both the implementation of the method and in the assessment of its worth. • even though ABC is often presented as a con ver ging method that approximates Bayesian inference, it can also be percei ved as an inference technique per se and hence analysed in its o wn right. Connections with indirect inference have already been drawn, ho we ver the fine asymptotic analysis of ABC w ould be most useful to derive. Moreov er , it could indirectly provide indications about the optimal calibration of the algorithm. • in connection with the above, the connection of ABC-based inference with other approxi- mati ve methods like variational Bayes inference is so far unexplored. Comparing and inter - breeding those dif ferent methods should become a research focus as well. • the construction and selection of the summary statistics is so far highly empirical. An auto- mated approach based on the principles of data analysis and approximate suf ficiency would be much more attracti ve and con vincing, especially in non-standard and complex settings. • the debate about ABC-based model choice is so far inconclusiv e in that we cannot guaran- tee the validity of the approximation, while considering that a “large enough” collection of summary statistics pro vides an acceptable level of approximation. Evaluating the discrep- ancy by exploratory methods like the bootstrap would shed a much more satisfactory light on this issue. • the method necessarily faces limitations imposed by large datasets or complex models, in that simulating pseudo-data may itself become an impossible task. Dimension-reducing technique that would simulate directly the summary statistics will quickly become neces- sary . Acknowledgements The authors are grateful to J.-M. Cornuet for bringing the problem to their attention and for a highly enjoyable and fruitful collaboration ov er the past years. Part of this w ork was conducted while the third author w as visiting the Department of Statistics at the Wharton Business School of the Univ ersity of Pennsylv ania, to whom he is most grateful for its support. The authors are also grateful to J.-L. Foule y for se veral interesting discussions. 24 Refer ences Bartolucci F , Scaccia L, Mira A (2006) Efficient Bayes factor estimation from the re versible jump output. 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