Approximate Bayesian Computation — TEI-tagged view

Mikael SunnåkerpersName, Alberto Giovanni BusettopersName, Elina NumminenpersName, Jukka CoranderpersName, Matthieu FollpersName, Christophe DessimozpersName (2013date). “Approximate Bayesian Computation.” PLoS Computational Biologytitle·j.

Object of study — approximate Bayesian computationterm·mth as a class of likelihood-free inference methods

Research question — how to do inference when the likelihoodterm·mth is elusive or too costly to evaluate

Contribution — a reference account of ABCterm·mth methods, their assumptions and their pitfalls

Full text — tagged

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Approximate Bayesian computationterm·mth (ABCterm·mth) constitutes a class of computational methodsterm·mth rooted in Bayesian statistics . In all model-based statistical inference , the likelihood functionterm·key is of central importance, since it expresses the probability of the observed data under a particular statistical model, and thus quantifies the support data lend to particular values of parameters and to choices among different models. For simple models, an analytical formula for the likelihood functionterm·key can typically be derived. However, for more complex models, an analytical formula might be elusive or the likelihood functionterm·key might be computationally very costly to evaluate. ABCterm·mth methods bypass the evaluation of the likelihood functionterm·key. In this way, ABCterm·mth methods widen the realm of models for which statistical inference can be considered. ABCterm·mth methods are mathematically well-founded, but they inevitably make assumptions and approximations whose impact needs to be carefully assessed. Furthermore, the wider application domain of ABCterm·mth exacerbates the challenges of parameter estimation and model selectionterm·key . ABCterm·mth has rapidly gained popularity over the last years and in particular for the analysis of complex problems arising in biological sciences (e.g., in population genetics , ecology , epidemiology , and systems biology ).

This is a “Topic Page” article for PLOS Computational Biology .

History The first Approximate Bayesian computationterm·mth (ABCterm·mth)-related ideas date back to the 1980s. Donald Rubin , when discussing the interpretation of Bayesian statements in 1984 [1] , described a hypothetical sampling mechanism that yields a sample from the posterior distributionterm·mth . This scheme was more of a conceptual thought experiment to demonstrate what type of manipulations are done when inferring the posterior distributionsterm·mth of parameters. The description of the sampling mechanism coincides exactly with that of the ABC-rejection scheme, and this article can be considered to be the first to describe approximate Bayesian computationterm·mth. Another prescient point was made when Rubin argued that in Bayesian inference, applied statisticians should not settle for analytically tractable models only but instead consider computational methodsterm·mth that allow them to estimate the posterior distributionterm·mth of interest. This way, a wider range of models can be considered. These arguments are particularly relevant in the context of ABCterm·mth.

In 1984, Peter Diggle and Richard Gratton suggested using a systematic simulation scheme to approximate the likelihood functionterm·key in situations where its analytic form is intractable [2] . Their method was based on defining a grid in the parameter space and using it to approximate the likelihoodterm·mth by running several simulations for each grid point. The approximation was then improved by applying smoothing techniques to the outcomes of the simulations. While the idea of using simulation for hypothesis testing was not new [3] , [4] , Diggle and Gratton seemingly introduced the first procedure using simulation to do statistical inference under a circumstance where the likelihoodterm·mth is intractable.

Although Diggle and Gratton's approach had opened a new frontier, their method was not yet exactly identical to what is now known as ABCterm·mth, as it aimed at approximating the likelihoodterm·mth rather than the posterior distributionterm·mth. An article of Simon Tavaré et al. [5] was first to propose an ABCterm·mth algorithm for posterior inference. In their seminal work, inference about the genealogy of DNA sequence data was considered, and in particular the problem of deciding the posterior distributionterm·mth of the time to the most recent common ancestor of the sampled individuals. Such inference is analytically intractable for many demographical models, but the authors presented ways of simulating coalescent trees under the putative models. A sample from the posterior of model parameters was obtained by accepting/rejecting proposals based on comparing the number of segregating sites in the synthetic and real data. This work was followed by an applied study on modeling the variation in human Y chromosome by Jonathan K. Pritchard et al. [6] using the ABCterm·mth method. Finally, the term Approximate Bayesian Computationterm·mth was established by Mark Beaumont et al. [7] , extending further the ABCterm·mth methodology and discussing the suitability of the ABC-approach more specifically for problems in population genetics. Since then, ABCterm·mth has spread to applications outside population genetics, such as systems biology, epidemiology, or phylogeography .

Method Motivation A common incarnation of the Bayes theorem relates the conditional probability (or density) of a particular parameter value given data to the probability of given by the rule: where denotes the posterior, the likelihoodterm·mth, the prior, and the evidence (also referred to as the marginal likelihoodterm·mth or the prior predictive probability of the data).

The prior represents beliefs about before is available, and it is often specified by choosing a particular distribution among a set of well-known and tractable families of distributions, such that both the evaluation of prior probabilities and random generation of values of are relatively straightforward. For certain kinds of models, it is more pragmatic to specify the prior using a factorization of the joint distribution of all the elements of in terms of a sequence of their conditional distributions. If one is only interested in the relative posterior plausibilities of different values of , the evidence can be ignored, as it constitutes a normalising constant , which cancels for any ratio of posterior probabilities. It remains, however, necessary to evaluate the likelihoodterm·mth and the prior . For numerous applications, it is computationally expensive , or even completely infeasible, to evaluate the likelihoodterm·mth [8] , which motivates the use of ABCterm·mth to circumvent this issue.

The ABCterm·mth Rejection Algorithmterm·key All ABC-based methods approximate the likelihood functionterm·key by simulations, the outcomes of which are compared with the observed data [9] – [11] . More specifically, with the ABCterm·mth rejection algorithmterm·key—the most basic form of ABCterm·mth—a set of parameter points is first sampled from the prior distribution. Given a sampled parameter point , a dataset is then simulated under the statistical model specified by . If the generated is too different from the observed data , the sampled parameter value is discarded. In precise terms, is accepted with toleranceterm·key if: where the distance measure determines the level of discrepancy between and based on a given metric (e.g., the Euclidean distance ). A strictly positive toleranceterm·key is usually necessary, since the probability that the simulation outcome coincides exactly with the data (event ) is negligible for all but trivial applications of ABCterm·mth, which would in practice lead to rejection of nearly all sampled parameter points. The outcome of the ABCterm·mth rejection algorithmterm·key is a sample of parameter values approximately distributed according to the desired posterior distributionterm·mth and, crucially, obtained without the need of explicitly evaluating the likelihood functionterm·key ( Figure 1 ).

Parameter estimation by Approximate Bayesian Computationterm·mth: a conceptual overview.

Summary Statisticsterm·mth The probability of generating a dataset with a small distance to typically decreases as the dimensionality of the data increases. This leads to a substantial decrease in the computational efficiency of the above basic ABCterm·mth rejection algorithmterm·key. A common approach to lessen this problem is to replace with a set of lower dimensional summary statisticsterm·mth , which are selected to capture the relevant information in . The acceptance criterion in ABCterm·mth rejection algorithmterm·key becomes: If the summary statisticsterm·mth are sufficient with respect to the model parameters , the efficiency increase obtained in this way does not introduce any error [12] . Indeed, by definition, sufficiency implies that all information in about is captured by .

As elaborated below, it is typically impossible, outside the exponential family of distributions , to identify a finite-dimensional set of sufficient statistics. Nevertheless, informative, but possibly nonsufficient, summary statisticsterm·mth are often used in applications where inference is performed with ABCterm·mth methods.

Example An illustrative example is a bistable system that can be characterized by a hidden Markov modelterm·mth (HMMterm·mth) subject to measurement noise ( Figure 2 ). Such models are employed for many biological systems: they have for example been used in Development , signaling , activation/deactivation , logical processing, and non-equilibrium thermodynamics . For instance, the behavior of the Sonic Hedgehog (Shh) transcription factor in Drosophila melanogaster can be modeled with a HMMterm·mth [13] . The (biological) dynamical model consists of two states: and . If the probability of a transition from one state to the other is defined as in both directions, the probability to remain in the same state at each time step is . The probability to measure the state correctly is (conversely, the probability of an incorrect measurement is ).

A dynamic bistable hidden Markov modelterm·mth.

Due to the conditional dependencies between states at different time points, calculation of the likelihoodterm·mth of time series data is somewhat tedious, which illustrates the motivation to use ABCterm·mth. A computational issue for the basic ABCterm·mth is the large dimensionality of the data in an application like this. This can be reduced using the summary statisticterm·mth , which is the frequency of switches between the two states. As a distance measure , the absolute difference is used, combined with a toleranceterm·key . The posterior inference about the parameter can be done following the five steps presented in Figure 1 : Step 1: Assume that the observed data are the state sequence AAAABAABBAAAAAABAAAA , which was generated using . The associated summary statisticterm·mth, the number of switches between the states in the experimental data, is .

Step 2: Assuming nothing is known about , a uniform prior in the interval is employed. A number of parameter points are drawn from the prior, and the model is simulated for each of the parameter points , which results in sequences of simulated data. In this example, , with each drawn parameter and simulated dataset recorded in Table 1 , column 2–3. In practice, would need to be much larger to obtain an appropriate approximation.

Example of ABCterm·mth rejection algorithmterm·key.

Step 3: The summary statisticterm·mth is being computed for each sequence of simulated data, ( Table 1 , column 4).

Step 4: The distance between the observed and simulated transition frequencies is computed for all parameter points ( Table 1 , column 5). Parameter points for which the distance is smaller than or equal to are accepted as approximate samples from the posterior ( Table 1 , column 6).

Step 5: The posterior distributionterm·mth is approximated with the accepted parameter points. The posterior distributionterm·mth should have a nonnegligible probability for parameter values in a region around the true value of in the system, if the data are sufficiently informative. In this example, the posterior probability mass is evenly split between the values and .

Figure 3 shows the posterior probabilities obtained by ABCterm·mth and a large using either the summary statisticterm·mth combined with ( and ) or the full data sequence. These are compared with the true posterior, which can be computed exactly and efficiently using the Viterbi algorithm . The used summary statisticterm·mth is not sufficient, and it is seen that even with , the deviation from the theoretical posterior is considerable. Of note, a much longer observed data sequence would be required to obtain a posterior that is concentrated around the true value of ( ).

Posterior of θ obtained in the example (red), compared with the true posterior distributionterm·mth (black), and ABCterm·mth simulations with large n . The use of the insufficient summary statisticterm·mth ω introduces a bias, even when requiring ε = 0 (light green).

This example application of ABCterm·mth used simplifications for illustrative purposes. A number of reviewterm·mth articles provide pointers to more realistic applications of ABCterm·mth [9] – [11] , [14] .

Model Comparison with ABCterm·mth Besides parameter estimation, the ABC-framework can be used to compute the posterior probabilities of different candidate models [15] – [17] . In such applications, one possibility is to use the rejection-sampling in a hierarchical manner. First, a model is sampled from the prior distribution for the models; then, given the model sampled, the model parameters are sampled from the prior distribution assigned to that model. Finally, a simulation is performed as in the single-model ABCterm·mth. The relative acceptance frequencies for the different models now approximate the posterior distributionterm·mth for these models. Again, computational improvements for ABCterm·mth in the space of models have been proposed, such as constructing a particle filter in the joint space of models and parameters [17] .

Once the posterior probabilities of models have been estimated, one can make full use of the techniques of Bayesian model comparison . For instance, to compare the relative plausibilities of two models and , one can compute their posterior ratio, which is related to the Bayes factorterm·key : If the model priors are equal ( ), the Bayes factorterm·key equals the posterior ratio.

In practice, as discussed below, these measures can be highly sensitive to the choice of parameter prior distributions and summary statisticsterm·mth, and thus conclusions of model comparison should be drawn with caution.

Pitfalls and Remedies As for all statistical methods, a number of assumptions and approximations are inherently required for the application of ABC-based methods to real modeling problems. For example, setting the toleranceterm·key parameter to zero ensures an exact result but typically makes computations prohibitively expensive. Thus, values of larger than zero are used in practice, which introduces a bias. Likewise, sufficient statistics are typically not available, and instead, other summary statisticsterm·mth are used, which introduces an additional bias due to the loss of information. Additional sources of bias—for example, in the context of model selectionterm·key—may be more subtle [12] , [18] .

At the same time, some of the criticisms that have been directed at the ABCterm·mth methods, in particular within the field of phylogeography [19] – [21] , are not specific to ABCterm·mth and apply to all Bayesian methods or even all statistical methods (e.g., the choice of prior distribution and parameter ranges) [9] , [22] . However, because of the ability of ABC-methods to handle much more complex models, some of these general pitfalls are of particular relevance in the context of ABCterm·mth analyses.

This section discusses these potential risks and reviewsterm·mth possible ways to address them ( Table 2 ).

Potential risks and remedies in ABC-based statistical inference.

Approximation of the Posterior A nonnegligible comes with the price that one samples from instead of the true posterior . With a sufficiently small toleranceterm·key, and a sensible distance measure, the resulting distribution should often approximate the actual target distribution reasonably well. On the other hand, a toleranceterm·key that is large enough that every point in the parameter space becomes accepted will yield a replica of the prior distribution. There are empirical studies of the difference between and as a function of [23] , and theoretical results for an upper -dependent bound for the error in parameter estimates [24] . The accuracy of the posterior (defined as the expected quadratic loss) delivered by ABCterm·mth as a function of has also been investigated [25] . However, the convergence of the distributions when approaches zero, and how it depends on the distance measure used, is an important topic that has yet to be investigated in greater detail. In particular, it remains difficult to disentangle errors introduced by this approximation from errors due to model mis-specification [9] .

As an attempt to correct some of the error due to a non-zero , the usage of local linear weighted regression with ABCterm·mth to reduce the variance of the posterior estimates has been suggested [7] . The method assigns weights to the parameters according to how well simulated summaries adhere to the observed ones and performs linear regression between the summaries and the weighted parameters in the vicinity of observed summaries. The obtained regression coefficients are used to correct sampled parameters in the direction of observed summaries. An improvement was suggested in the form of nonlinear regression using a feed-forward neural network model [26] . However, it has been shown that the posterior distributionsterm·mth obtained with these approaches are not always consistent with the prior distribution, which did lead to a reformulation of the regression adjustment that respects the prior distribution [27] .

Finally, statistical inference using ABCterm·mth with a non-zero toleranceterm·key is not inherently flawed: under the assumption of measurement errors, the optimal can in fact be shown to be not zero [25] , [28] . Indeed, the bias caused by a non-zero toleranceterm·key can be characterized and compensated by introducing a specific form of noise to the summary statisticsterm·mth. Asymptotic consistency for such “noisy ABCterm·mth” has been established, together with formulas for the asymptotic variance of the parameter estimates for a fixed toleranceterm·key [25] .

Choice and Sufficiency of Summary Statisticsterm·mth Summary statisticsterm·mth may be used to increase the acceptance rate of ABCterm·mth for high-dimensional data. Low-dimensional sufficient statistics are optimal for this purpose, as they capture all relevant information present in the data in the simplest possible form [11] . However, low-dimensional sufficient statistics are typically unattainable for statistical models where ABC-based inference is most relevant, and consequently, some heuristic is usually necessary to identify useful low-dimensional summary statisticsterm·mth. The use of a set of poorly chosen summary statisticsterm·mth will often lead to inflated credible intervals due to the implied loss of information [11] , which can also bias the discrimination between models. A reviewterm·mth of methods for choosing summary statisticsterm·mth is available [29] , which may provide valuable guidance in practice.

One approach to capture most of the information present in data would be to use many statistics, but the accuracy and stability of ABCterm·mth appears to decrease rapidly with an increasing numbers of summary statisticsterm·mth [9] , [11] . Instead, a better strategy is to focus on the relevant statistics only—relevancy depending on the whole inference problem, on the model used, and on the data at hand [30] .

An algorithm has been proposed for identifying a representative subset of summary statisticsterm·mth, by iteratively assessing whether an additional statistic introduces a meaningful modification of the posterior [31] . One of the challenges here is that a large ABCterm·mth approximation error may heavily influence the conclusions about the usefulness of a statistic at any stage of the procedure. Another method decomposes into two main steps [30] . First, a reference approximation of the posterior is constructed by minimizing the entropy . Sets of candidate summaries are then evaluated by comparing the ABC-approximated posteriors with the reference posterior.

With both of these strategies, a subset of statistics is selected from a large set of candidate statistics. Instead, the partial least squares regression approach uses information from all the candidate statistics, each being weighted appropriately [32] . Recently, a method for constructing summaries in a semi-automatic manner has attained a considerable interest [25] . This method is based on the observation that the optimal choice of summary statisticsterm·mth, when minimizing the quadratic loss of the parameter point estimates, can be obtained through the posterior mean of the parameters, which is approximated by performing a linear regression based on the simulated data.

Methods for the identification of summary statisticsterm·mth that could also simultaneously assess the influence on the approximation of the posterior would be of substantial value [33] . This is because the choice of summary statisticsterm·mth and the choice of toleranceterm·key constitute two sources of error in the resulting posterior distributionterm·mth. These errors may corrupt the ranking of models and may also lead to incorrect model predictions. Indeed, none of the methods above assess the choice of summaries for the purpose of model selectionterm·key.

Bayes Factorterm·key with ABCterm·mth and Summary Statisticsterm·mth It has been shown that the combination of insufficient summary statisticsterm·mth and ABCterm·mth for model selectionterm·key can be problematic [12] , [18] . Indeed, if one lets the Bayes factorterm·key based on the summary statisticterm·mth be denoted by , the relation between and takes the form [12] : Thus, a summary statisticterm·mth is sufficient for comparing two models and if and only if: which results in that . It is also clear from the equation above that there might be a huge difference between and if the condition is not satisfied, as can be demonstrated by toy examples [12] , [16] , [18] . Crucially, it was shown that sufficiency for or alone, or for both models, does not guarantee sufficiency for ranking the models [12] . However, it was also shown that any sufficient summary statisticterm·mth for a model in which both and are nested is valid for ranking the nested models [12] .

The computation of Bayes factorsterm·key on may therefore be misleading for model selectionterm·key purposes, unless the ratio between the Bayes factorsterm·key on and would be available, or at least could be approximated reasonably well. Alternatively, necessary and sufficient conditions on summary statisticsterm·mth for a consistent Bayesian model choice have recently been derived [34] , which can provide useful guidance.

However, this issue is only relevant for model selectionterm·key when the dimension of the data has been reduced. ABC-based inference, in which the actual datasets are directly compared—as is the case for some systems biology applications (e.g., see [35] )—circumvents this problem.

Indispensable Quality Controls As the above discussion makes clear, any ABCterm·mth analysis requires choices and tradeoffs that can have a considerable impact on its outcomes. Specifically, the choice of competing models/hypotheses, the number of simulations, the choice of summary statisticsterm·mth, or the acceptance threshold cannot currently be based on general rules, but the effect of these choices should be evaluated and tested in each study [10] .

A number of heuristic approaches to the quality control of ABCterm·mth have been proposed, such as the quantification of the fraction of parameter variance explained by the summary statisticsterm·mth [10] . A common class of methods aims at assessing whether or not the inference yields valid results, regardless of the actually observed data. For instance, given a set of parameter values, which are typically drawn from the prior or the posterior distributionsterm·mth for a model, one can generate a large number of artificial datasets. In this way, the quality and robustness of ABCterm·mth inference can be assessed in a controlled setting, by gauging how well the chosen ABCterm·mth inference method recovers the true parameter values, and also models if multiple structurally different models are considered simultaneously.

Another class of methods assesses whether the inference was successful in light of the given observed data, for example by comparing the posterior predictive distribution of summary statisticsterm·mth to the summary statisticsterm·mth observed [10] . Beyond that, cross-validation techniques [36] and predictive checks [37] , [38] represent promising future strategies to evaluate the stability and out-of-sample predictive validity of ABCterm·mth inferences. This is particularly important when modeling large datasets, because then the posterior support of a particular model can appear overwhelmingly conclusive, even if all proposed models in fact are poor representations of the stochastic system underlying the observation data. Out-of-sample predictive checks can reveal potential systematic biases within a model and provide clues on to how to improve its structure or parametrization.

Interestingly, fundamentally novel approaches for model choice that incorporate quality control as an integral step in the process have recently been proposed. ABCterm·mth allows, by construction, estimation of the discrepancies between the observed data and the model predictions, with respect to a comprehensive set of statistics. These statistics are not necessarily the same as those used in the acceptance criterion. The resulting discrepancy distributions have been used for selecting models that are in agreement with many aspects of the data simultaneously [39] , and model inconsistency is detected from conflicting and codependent summaries. Another quality-control-based method for model selectionterm·key employs ABCterm·mth to approximate the effective number of model parameters and the deviance of the posterior predictive distributions of summaries and parameters [40] . The deviance information criterion is then used as measure of model fit. It has also been shown that the models preferred based on this criterion can conflict with those supported by Bayes factorsterm·key . For this reason, it is useful to combine different methods for model selectionterm·key to obtain correct conclusions.

Quality controls are achievable and indeed performed in many ABC-based works, but for certain problems, the assessment of the impact of the method-related parameters can be challenging. However, the rapidly increasing use of ABCterm·mth can be expected to provide a more thorough understanding of the limitations and applicability of the method.

General Risks in Statistical Inference Exacerbated in ABCterm·mth This section reviewsterm·mth risks that are strictly speaking not specific to ABCterm·mth, but also relevant for other statistical methods as well. However, the flexibility offered by ABCterm·mth to analyze very complex models makes them highly relevant to discuss here.

Prior distribution and parameter ranges The specification of the range and the prior distribution of parameters strongly benefits from previous knowledge about the properties of the system. One criticism has been that in some studies the “parameter ranges and distributions are only guessed based upon the subjective opinion of the investigators” [41] , which is connected to classical objections of Bayesian approaches [42] .

With any computational methodterm·mth, it is typically necessary to constrain the investigated parameter ranges. The parameter ranges should if possible be defined based on known properties of the studied system but may for practical applications necessitate an educated guess. However, theoretical results regarding objective priors are available, which may for example be based on the principle of indifference or the principle of maximum entropy [43] , [44] . On the other hand, automated or semi-automated methods for choosing a prior distribution often yield improper densities . As most ABCterm·mth procedures require generating samples from the prior, improper priors are not directly applicable to ABCterm·mth.

One should also keep the purpose of the analysis in mind when choosing the prior distribution. In principle, uninformative and flat priors that exaggerate our subjective ignorance about the parameters may still yield reasonable parameter estimates. However, Bayes factorsterm·key are highly sensitive to the prior distribution of parameters. Conclusions on model choice based on Bayes factorterm·key can be misleading unless the sensitivity of conclusions to the choice of priors is carefully considered.

Small number of models Model-based methods have been criticized for not exhaustively covering the hypothesis space [21] . Indeed, model-based studies often revolve around a small number of models, and due to the high computational cost to evaluate a single model in some instances, it may then be difficult to cover a large part of the hypothesis space.

An upper limit to the number of considered candidate models is typically set by the substantial effort required to define the models and to choose between many alternative options [10] . There is no commonly accepted ABC-specific procedure for model construction, so experience and prior knowledge are used instead [11] . Although more robust procedures for a priori model choice and formulation would be beneficial, there is no one-size-fits-all strategy for model development in statistics: sensible characterization of complex systems will always necessitate a great deal of detective work and use of expert knowledge from the problem domain.

Some opponents of ABCterm·mth contend that since only few models—subjectively chosen and probably all wrong—can be realistically considered, ABCterm·mth analyses provide only limited insight [21] . However, there is an important distinction between identifying a plausible null hypothesis and assessing the relative fit of alternative hypotheses [9] . Since useful null hypotheses, that potentially hold true, can extremely seldom be put forward in the context of complex models, predictive ability of statistical models as explanations of complex phenomena is far more important than the test of a statistical null hypothesis in this context. It is also common to average over the investigated models, weighted based on their relative plausibility, to infer model features (e.g., parameter values) and to make predictions.

Large datasets Large datasets may constitute a computational bottleneck for model-based methods. It was, for example, pointed out that in some ABC-based analyses, part of the data have to be omitted [21] . A number of authors have argued that large datasets are not a practical limitation [10] , [42] , although the severity of this issue depends strongly on the characteristics of the models. Several aspects of a modeling problem can contribute to the computational complexity, such as the sample size, number of observed variables or features, time or spatial resolution, etc. However, with increasing computing power, this issue will potentially be less important.

Instead of sampling parameters for each simulation from the prior, it has been proposed alternatively to combine the Metropolis-Hastings algorithm with ABCterm·mth, which was reported to result in a higher acceptance rate than for plain ABCterm·mth [33] . Naturally, such an approach inherits the general burdens of MCMC methods, such as the difficulty to assess convergence, correlation among the samples from the posterior [23] , and relatively poor parallelizability [10] .

Likewise, the ideas of sequential Monte Carloterm·key (SMC) and population Monte Carlo (PMC) methods have been adapted to the ABCterm·mth setting [23] , [45] . The general idea is to iteratively approach the posterior from the prior through a sequence of target distributions. An advantage of such methods, compared to ABC-MCMC, is that the samples from the resulting posterior are independent. In addition, with sequential methods the toleranceterm·key levels must not be specified prior to the analysis, but are adjusted adaptively [46] .

It is relatively straightforward to parallelize a number of steps in ABCterm·mth algorithms based on rejection sampling and sequential Monte Carloterm·key methods. It has also been demonstrated that parallel algorithms may yield significant speedups for MCMC-based inference in phylogenetics [47] , which may be a tractable approach also for ABC-based methods. Yet an adequate model for a complex system is very likely to require intensive computation irrespectively of the chosen method of inference, and it is up to the user to select a method that is suitable for the particular application in question.

Curse-of-dimensionalityterm·key High-dimensional datasets and high-dimensional parameter spaces can require an extremely large number of parameter points to be simulated in ABC-based studies to obtain a reasonable level of accuracy for the posterior inferences. In such situations, the computational cost is severely increased and may in the worst case render the computational analysis intractable. These are examples of well-known phenomena, which are usually referred to with the umbrella term curse-of-dimensionalityterm·key [48] .

To assess how severely the dimensionality of a dataset affects the analysis within the context of ABCterm·mth, analytical formulas have been derived for the error of the ABCterm·mth estimators as functions of the dimension of the summary statisticsterm·mth [49] – [50] . In addition, Blum and François have investigated how the dimension of the summary statisticsterm·mth is related to the mean squared error for different correction adjustments to the error of ABCterm·mth estimators. It was also argued that dimension reduction techniques are useful to avoid the curse-of-dimensionalityterm·key, due to a potentially lower dimensional underlying structure of summary statisticsterm·mth [49] . Motivated by minimizing the quadratic loss of ABCterm·mth estimators, Fearnhead and Prangle have proposed a scheme to project (possibly high-dimensional) data into estimates of the parameter posterior means; these means, now having the same dimension as the parameters, are then used as summary statisticsterm·mth for ABCterm·mth [50] .

ABCterm·mth can be used to infer problems in high-dimensional parameter spaces, although one should account for the possibility of overfitting (e.g., see the model selectionterm·key methods in [39] – [40] ). However, the probability of acceptingterm·mth the simulated values for the parameters under a given toleranceterm·key with the ABCterm·mth rejection algorithmterm·key typically decreases exponentially with increasing dimensionality of the parameter spaceterm·mth (due to the global acceptance criterion) [11] . Although no computational methodterm·mth (based on ABCterm·mth or not) seems to be able to break the curse-of-dimensionalityterm·key, methods have recently been developed to handle high-dimensional parameter spaces under certain assumptions (e.g., based on polynomial approximation on sparse grids [51] , which could potentially heavily reduce the simulation times for ABCterm·mth). However, the applicability of such methods is problem dependent, and the difficulty of exploring parameter spaces should in general not be underestimated. For example, the introduction of deterministic global parameter estimation led to reports that the global optima obtained in several previous studies of low-dimensional problems were incorrect [52] . For certain problems, it might therefore be difficult to know whether the model is incorrect or, as discussed above, whether the explored region of the parameter space is inappropriate [21] . A more pragmatic approach is to cut the scope of the problem through model reduction [11] .

Software for Application of ABCterm·mth A number of software packages are currently available for application of ABCterm·mth to particular classes of statistical models. An assortment of ABC-based software is presented in Table 3 .

Software incorporating ABCterm·mth.

The suitability of individual software packages is dependent on the specific application at hand, the computer system environment, and the algorithms required.

Conclusion In conclusion, ABCterm·mth represents a class of well-founded and powerful methods for Bayesian statistical inference. However, reliable application of ABCterm·mth requires additional caution to be considered, due to the approximations and biases introduced at the different stages of the approach. In its current incarnation, the ABCterm·mth toolkit as a whole is best suited for inference about parameters or predictive inferences about observables in the presence of a single or few candidate model(s). How to make ABCterm·mth practically feasible for problems involving large sets of models and/or high-dimensional target parameter spaces is currently largely an open issue. Since the computation of the likelihood functionterm·key is bypassed, it can be tempting to attack high-dimensional problems using ABCterm·mth, but inevitably this comes bundled with new challenges that investigators need to be aware of at each step of their analyses.

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