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Although Diggle and Gratton's approach had opened a new frontier, their method was not yet exactly identical to what is now known as ABC, as it aimed at approximating the likelihood rather than the posterior distribution. An article of [[Simon Tavaré]] and co-authors was first to propose an ABC algorithm for posterior inference.<ref name="Tavare" /> In their seminal work, inference about the genealogy of DNA sequence data was considered, and in particular the problem of deciding the posterior distribution of the time to the [[most recent common ancestor]] of the sampled individuals. Such inference is analytically intractable for many demographic 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]] and co-authors using the ABC method.<ref name="Pritchard1999" /> Finally, the term approximate Bayesian computation was established by Mark Beaumont and co-authors,<ref name="Beaumont2002" /> extending further the ABC methodology and discussing the suitability of the ABC-approach more specifically for problems in population genetics. Since then, ABC has spread to applications outside population genetics, such as systems biology, epidemiology, and [[phylogeography]].
Approximate Bayesian computation can be understood as a kind of Bayesian version of [[indirect inference]].<ref>
Several efficient Monte Carlo based approaches have been developed to perform sampling from the ABC posterior distribution for purposes of estimation and prediction problems. A popular choice is the SMC Samplers algorithim <ref>{{Cite journal |last1=Del Moral |first1=Pierre |last2=Doucet |first2=Arnaud |last3=Jasra |first3=Ajay |date=2006 |title=Sequential Monte Carlo Samplers |url=https://www.jstor.org/stable/3879283 |journal=Journal of the Royal Statistical Society. Series B (Statistical Methodology) |volume=68 |issue=3 |pages=411–436 |doi=10.1111/j.1467-9868.2006.00553.x |jstor=3879283 |issn=1369-7412|arxiv=cond-mat/0212648 }}</ref><ref>{{Cite journal |last1=Del Moral |first1=Pierre |last2=Doucet |first2=Arnaud |last3=Peters |first3=Gareth |date=2004 |title=Sequential Monte Carlo Samplers CUED Technical Report |url=https://www.ssrn.com/abstract=3841065 |journal=SSRN Electronic Journal |language=en |doi=10.2139/ssrn.3841065 |issn=1556-5068}}</ref><ref>{{Cite journal |last=Peters |first=Gareth |date=2005 |title=Topics in Sequential Monte Carlo Samplers |url=https://www.ssrn.com/abstract=3785582 |journal=SSRN Electronic Journal |language=en |doi=10.2139/ssrn.3785582 |issn=1556-5068}}</ref> adapted to the ABC context in the method (SMC-ABC).<ref>{{Cite journal |last1=Sisson |first1=S. A. |last2=Fan |first2=Y. |last3=Tanaka |first3=Mark M. |date=2007-02-06 |title=Sequential Monte Carlo without likelihoods |journal=Proceedings of the National Academy of Sciences |language=en |volume=104 |issue=6 |pages=1760–1765 |doi=10.1073/pnas.0607208104 |doi-access=free |issn=0027-8424 |pmc=1794282 |pmid=17264216|bibcode=2007PNAS..104.1760S }}</ref><ref>{{Cite journal |last=Peters |first=Gareth |date=2009 |title=Advances in Approximate Bayesian Computation and Trans-Dimensional Sampling Methodology |url=https://www.ssrn.com/abstract=3785580 |journal=SSRN Electronic Journal |language=en |doi=10.2139/ssrn.3785580 |issn=1556-5068}}</ref><ref>{{Cite journal |last1=Peters |first1=G. W. |last2=Sisson |first2=S. A. |last3=Fan |first3=Y. |date=2012-11-01 |title=Likelihood-free Bayesian inference for α-stable models |url=https://www.sciencedirect.com/science/article/pii/S0167947310003786 |journal=Computational Statistics & Data Analysis |series=1st issue of the Annals of Computational and Financial Econometrics |volume=56 |issue=11 |pages=3743–3756 |doi=10.1016/j.csda.2010.10.004 |issn=0167-9473}}</ref><ref>{{Cite journal |last1=Peters |first1=Gareth W. |last2=Wüthrich |first2=Mario V. |last3=Shevchenko |first3=Pavel V. |date=2010-08-01 |title=Chain ladder method: Bayesian bootstrap versus classical bootstrap |url=https://www.sciencedirect.com/science/article/pii/S0167668710000351 |journal=Insurance: Mathematics and Economics |volume=47 |issue=1 |pages=36–51 |doi=10.1016/j.insmatheco.2010.03.007 |arxiv=1004.2548 |issn=0167-6687}}</ref>
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===Choice and sufficiency of summary statistics===
Summary statistics may be used to increase the acceptance rate of ABC 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.<ref name="Csillery" /><ref>{{Cite journal |last1=Peters |first1=Gareth William |last2=Wuthrich |first2=Mario V. |last3=Shevchenko |first3=Pavel V. |date=2009 |title=Chain Ladder Method: Bayesian Bootstrap Versus Classical Bootstrap |url=https://dx.doi.org/10.2139/ssrn.2980411 |journal=SSRN Electronic Journal |doi=10.2139/ssrn.2980411 |arxiv=1004.2548 |issn=1556-5068}}</ref><ref>{{
One approach to capture most of the information present in data would be to use many statistics, but the accuracy and stability of ABC appears to decrease rapidly with an increasing numbers of summary statistics.<ref name="Beaumont2010" /><ref name="Csillery" /> 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.<ref name="Nunes" />
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<ref name="Klinger2017">Klinger, E.; Rickert, D.; Hasenauer, J. (2017). pyABC: distributed, likelihood-free inference.</ref>
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