  Approximate Bayesian computation has emerged as a standard computational tool
when dealing with the increasingly common scenario of completely intractable
likelihood functions in Bayesian inference. We show that many common Markov
chain Monte Carlo kernels used to facilitate inference in this setting can fail
to be variance bounding, and hence geometrically ergodic, which can have
consequences for the reliability of estimates in practice. This phenomenon is
typically independent of the choice of tolerance in the approximation. We then
prove that a recently introduced Markov kernel in this setting can inherit
variance bounding and geometric ergodicity from its intractable
Metropolis--Hastings counterpart, under reasonably weak and manageable
conditions. We show that the computational cost of this alternative kernel is
bounded whenever the prior is proper, and present indicative results on an
example where spectral gaps and asymptotic variances can be computed, as well
as an example involving inference for a partially and discretely observed,
time-homogeneous, pure jump Markov process. We also supply two general
theorems, one of which provides a simple sufficient condition for lack of
variance bounding for reversible kernels and the other provides a positive
result concerning inheritance of variance bounding and geometric ergodicity for
mixtures of reversible kernels.
