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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 |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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