Review
Approximate Bayesian Computation (ABC) in practice

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Understanding the forces that influence natural variation within and among populations has been a major objective of evolutionary biologists for decades. Motivated by the growth in computational power and data complexity, modern approaches to this question make intensive use of simulation methods. Approximate Bayesian Computation (ABC) is one of these methods. Here we review the foundations of ABC, its recent algorithmic developments, and its applications in evolutionary biology and ecology. We argue that the use of ABC should incorporate all aspects of Bayesian data analysis: formulation, fitting, and improvement of a model. ABC can be a powerful tool to make inferences with complex models if these principles are carefully applied.

Section snippets

Inference with simulations in evolutionary genetics

Natural populations have complex demographic histories: their sizes and ranges change over time, leading to fission and fusion processes that leave signatures on their genetic composition [1]. One ‘promise’ of biology is that molecular data will help us uncover the complex demographic and adaptive processes that have acted on natural populations. The widespread availability of different molecular markers and increased computer power has fostered the development of sophisticated statistical

The ABC of Approximate Bayesian Computation

ABC has its roots in the rejection algorithm, a simple technique to generate samples from a probability distribution 8, 9. The basic rejection algorithm consists of simulating large numbers of datasets under a hypothesized evolutionary scenario. The parameters of the scenario are not chosen deterministically, but sampled from a probability distribution. The data generated by simulation are then reduced to summary statistics, and the sampled parameters are accepted or rejected on the basis of

Bayesian data analysis: building, fitting, and improving the model

The three main steps of Bayesian analysis are formulating the model, fitting the model to data, and improving the model by checking its fit and comparing it with other models [59] (Box 4). When formulating a model, we use our experience and prior knowledge, and sometimes resort to established theories. This step is often mathematical in essence because it encompasses explicit definitions for the likelihood (or a generating mechanism) and the prior distribution. These quantities summarize

Choice and dimension of summary statistics

Carrying out inference based on summary statistics instead of the full dataset inevitably implies discarding potentially useful information. More specifically, if a summary statistic is not sufficient for the parameter of interest, the posterior distribution computed with this statistic would not be equal to the posterior distribution computed with the full dataset [4]. Many areas of evolutionary biology focus on developing informative statistics. An example of a recently developed statistic is

Inference under complex models

ABC is extremely flexible and relatively easy to implement, so inference can be carried out for many complex models in evolution and ecology as long as informative summary statistics are available and simulating data under the model is possible. Templeton 56, 57 argues that the interpretation of why a complex model is preferred is highly subjective when using computer simulations. With the proliferation of highly complex models, Bayesian statisticians and evolutionary biologists have made

Conclusions

Biology is a complex science, so it is inevitable that the observation of biological systems leads us to build complex models. However, the apparent ease of using an inference algorithm such as ABC should never hide the general difficulties of making inferences under complex models. The automatic process of inference is hampered by the model definition and model checking steps, which are case-dependent and highly user-interactive. ABC is far from being as ‘easy as 123’. Important caveats when

Acknowledgements

KC is funded by a postdoctoral fellowship from the Université Joseph Fourier (ABC MSTIC). OF was partially supported by a grant from the Agence Nationale de la Recherche (BLAN06-3146282 MAEV). MB and OF acknowledge the support of the Complex Systems Institute (IXXI). OEG acknowledges support from the EcoChange Project.

Glossary

Bayes factor
the ratio of probabilities of two models that is used to evaluate the relative support of one model in relation to another in Bayesian model comparison.
Bayesian statistics
a general framework for summarizing uncertainty (prior information) and making estimates and predictions using probability statements conditional on observed data and an assumed model.
Coalescent theory
a mathematical theory that describes the ancestral relationships of a sample of ‘individuals’ back to their common

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