Asymptotic Behaviour of the Posterior Distribution in Approximate Bayesian Computation

Approximate Bayesian computation (ABC) is a popular technique for approximating likelihoods and is often used in parameter estimation when the likelihood functions are analytically intractable. In the context of Hidden Markov Models (HMMs), we analyse the asymptotic behaviour of the posterior distribution in ABC based Bayesian parameter estimation. In particular we show that Bernstein-von Mises type results still hold but that the resulting posterior is biased in the sense that it concentrates around a point in parameter space that differs from the true parameter value. Furthermore we obtain precise rates for the size of this bias with respect to a natural accuracy parameter of the ABC method. Finally we discuss, via a numerical example, the implications of our results for the practical implementation of ABC.