Approximate Bayesian Computation (ABC) is used in problems when the likelihood function is intractable by producing datasets that are sufficiently similar to the observed dataset

ABC is a computational technique for approximately simulating from a posterior distribution that proves unmanageable by analytic and regular simulation approaches, including MCMC. The technique can also be interpreted as a crude Bayesian non-parametric approach.

Some simple examples are provided in this entry. In such cases, it may prove impossible to run a regular Monte Carlo or Markov chain Monte Carlo approach. ABC relies instead on the assumed possibility to produce new samples from the same distribution as the data, $f(x|\theta)$ say, given a value $\theta$ of the parameter. It proceeds as follows: given a dataset $y^\text{obs}$, a model $f(x|\theta)$, and a prior $\pi(\theta)$,

  1. Generate $\theta_1,\ldots,\theta_N\stackrel{\text{iid}}{\sim}\pi(\theta)$;
  2. For each $\theta_i$, generate $z_i\sim f(z|\theta_i)$;
  3. Keep the $K$ $\theta_i$'s corresponding to the smallest distances $d(z_i,y^\text{obs})$

where the distance between the data and the simulated data usually depends on summaries of those. The outcome of this ABC algorithm is a simulation from the approximate posterior $$\pi(\theta|d(z,y)\le K/N)$$ The Wikipedia entry on ABC is quite informative and authoritative on the topic.