  Likelihood-free methods such as approximate Bayesian computation (ABC) have
extended the reach of statistical inference to problems with computationally
intractable likelihoods. Such approaches perform well for small-to-moderate
dimensional problems, but suffer a curse of dimensionality in the number of
model parameters. We introduce a likelihood-free approximate Gibbs sampler that
naturally circumvents the dimensionality issue by focusing on lower-dimensional
conditional distributions. These distributions are estimated by flexible
regression models either before the sampler is run, or adaptively during
sampler implementation. As a result, and in comparison to Metropolis-Hastings
based approaches, we are able to fit substantially more challenging statistical
models than would otherwise be possible. We demonstrate the sampler's
performance via two simulated examples, and a real analysis of Airbnb rental
prices using a intractable high-dimensional multivariate non-linear state space
model containing 13,140 parameters, which presents a real challenge to standard
ABC techniques.
