All ABC-based methods approximate the likelihood function by simulations, the outcomes of which are compared with the observed data.<ref name="Beaumont2010" /><ref name="Bertorelle" /><ref name="Csillery" /> More specifically, with the ABC rejection algorithm—thealgorithm — the most basic form of ABC—aABC — 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:
