Revisiting Concentration Results for Approximate Bayesian Computation

We study the large sample behavior of approximate Bayesian computation (ABC) posterior measures in situations when the data generating process is dependent on unidentifiable parameters. In particular, we establish the concentration of posterior measures on sets of arbitrarily small measure that contain the equivalence set of the data generative parameter, when the sample size tends to infinity. Our theory also makes weak assumptions regarding the measurement of discrepancy between the data set and simulations. In particular, it does not require the use of summary statistics and is applicable to a broad class of kernelized ABC algorithms. We provide useful illustrations and demonstrations of our theory in practice, and offer a comprehensive assessment of how our findings complement other results in the literature.