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I am trying to understand the link between (rejection) ABC and rejection sampling. For example, this paper states:

Approximate Bayesian Computation (ABC, Sisson et al., 2018) is centered around the idea of Monte Carlo rejection sampling (Tavaré et al., 1997; Pritchard et al., 1999). Parameters $\theta$ are sampled from a proposal distribution, simulation outcomes $x$ are compared with observed data $x_{o}$, and are accepted or rejected depending on a (user-specified) distance function and rejection criterion. While rejection ABC (REJ-ABC) uses the prior as a proposal distribution, [...]

Usually, the comparison of simulated data $x$ and observed data $x_{o}$ happens by choosing a threshold $\epsilon$ and a distance measure $\rho$, and if $\rho(x, x_{o}) < \epsilon$, then $\theta$ is accepted. Hence, I do not see any connection (need) for rejection sampling, as is stated in the quote. Can anybody please clarify?

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The basic ABC algorithm is based on a while loop:

While$$\rho(x^\text{obs},x^{sim})>\epsilon,\tag{1}$$simulate$$(\theta,x^\text{sim})\sim \pi(\theta)f(x^\text{sim}|\theta)$$

This means that the simulations from an instrumental distribution are rejected while (1) holds. This is a rudimentary form of (acceptance-) rejection sampling that does not require an extra uniform variate since the acceptance probability is either $0$ or $1$.

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