A Pseudo-Marginal Perspective on the ABC Algorithm

We make two observations about the efficiency of approximate Bayesian computation (ABC). First, we show that the MCMC version of an ABC algorithm is no more efficient than the corresponding MCMC algorithm. Thus likelihood-free MCMC methods should not be used if the corresponding MCMC algorithm is feasible to implement. Second, we observe that some variations of ABC algorithms can be viewed as pseudo-marginal MCMC algorithms, and hence may be made arbitrarily close to their respective likelihood-based MCMC methods. We analyze the efficiency of the resulting algorithm, and present a surprising result showing that multiple pseudo-samples do not necessarily improve the efficiency of the algorithm as compared to employing a high-variance estimate computed using a single pseudo-sample. Lastly, we demonstrate a case where multiple samples can provide benefit, specifically when subsequent samples have a reduced cost and the ABC bandwidth is small relative to the likelihood variance.