geometric ergodicity of ABC-MCMC

During a discussion with Natesh Pillai at ICERM, he pointed out to me the recent paper by Anthony Lee and Krzysztof Latuszynski on the (lack of) geometric ergodicity of ABC-MCMC algorithms. Since I had missed this paper, I went back and read it in the early hours…

The paper compares four algorithms, from the standard (#1) ABC-MCMC (with N replicates of the pseudo-data) to versions involving simulations of those replicates repeated at the subsequent step (#2), use of a stopping rule in the generation of the pseudo data (#3), and an “ideal” algorithm based on the (unavailable) measure of the ε ball around the data (#4). They recall a result by Tweedie and Roberts (1996), also used in Mengersen and Tweedie (1996), namely that the chain cannot be geometrically ergodic when there exist almost absorbing/sticky states. From there, they derive that (under their technical assumptions) versions #1 and #2 cannot be variance bounding (i.e. the spectral gap is zero), while #3 and #4 can.be both variance bounding and geometrically ergodici under conditions on the prior and the above ball measure. It is thus interesting if a wee mysterious that simulating a random number of auxiliary variables is sufficient to achieve geometric ergodicity.