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Approximate Bayesian computation can be understood as a kind of Bayesian version of [[indirect inference]].<ref>Drovandi, Christopher C. "ABC and indirect inference." Handbook of Approximate Bayesian Computation (2018): 179-209. https://arxiv.org/abs/1803.01999</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>
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 |
==Method==
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===The ABC rejection algorithm===
All ABC-based methods approximate the likelihood function by simulations, the outcomes of which are compared with the observed data.<ref>{{Cite journal |last=Hunter |first=Dawn |date=2006-12-08 |title=Bayesian inference, Monte Carlo sampling and operational risk |url=https://www.risk.net/journal-of-operational-risk/2160915/bayesian-inference-monte-carlo-sampling-and-operational-risk |journal=Journal of Operational Risk |volume=1 |issue=3 |pages=27–50 |language=en |doi=10.21314/jop.2006.014}}</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 name="Beaumont2010" /><ref name="Bertorelle" /><ref name="Csillery" /> More specifically, with the ABC rejection algorithm — the most basic form of ABC — a set of parameter points is first sampled from the prior distribution. Given a sampled parameter point <math>\hat{\theta}</math>, a data set <math>\hat{D}</math> is then simulated under the statistical model <math>M</math> specified by <math>\hat{\theta}</math>. If the generated <math>\hat{D}</math> is too different from the observed data <math>D</math>, the sampled parameter value is discarded. In precise terms, <math>\hat{D}</math> is accepted with tolerance <math>\epsilon \ge 0</math> if:
:<math>\rho (\hat{D},D)\le\epsilon</math>,
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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 |
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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