ABC with Gibbs steps

With Grégoire Clarté, Robin Ryder and Julien Stoehr, all from Paris-Dauphine, we have just arXived a paper on the specifics of ABC-Gibbs, which is a version of ABC where the generic ABC accept-reject step is replaced by a sequence of n conditional ABC accept-reject steps, each aiming at an ABC version of a conditional distribution extracted from the joint and intractable target. Hence an ABC version of the standard Gibbs sampler. What makes it so special is that each conditional can (and should) be conditioning on a different statistic in order to decrease the dimension of this statistic, ideally down to the dimension of the corresponding component of the parameter. This successfully bypasses the curse of dimensionality but immediately meets with two difficulties. The first one is that the resulting sequence of conditionals is not coherent, since it is not a Gibbs sampler on the ABC target. The conditionals are thus incompatible and therefore convergence of the associated Markov chain becomes an issue. We produce sufficient conditions for the Gibbs sampler to converge to a stationary distribution using incompatible conditionals. The second problem is then that, provided it exists, the limiting and also intractable distribution does not enjoy a Bayesian interpretation, hence may fail to be justified from an inferential viewpoint. We however succeed in producing a version of ABC-Gibbs in a hierarchical model where the limiting distribution can be explicited and even better can be weighted towards recovering the original target. (At least with limiting zero tolerance.)