Title: Variance bounding of delayed-acceptance kernels,
Abstract: A delayed-acceptance version of a Metropolis--Hastings algorithm can be
useful for Bayesian inference when it is computationally expensive to calculate
the true posterior, but a computationally cheap approximation is available; the
delayed-acceptance kernel targets the same posterior as its parent
Metropolis-Hastings kernel. Although the asymptotic variance of any functional
of the chain cannot be less than that obtained using its parent, the average
computational time per iteration can be much smaller and so for a given
computational budget the delayed-acceptance kernel can be more efficient.
When the asymptotic variance of all $L^2$ functionals of the chain is finite,
the kernel is said to be variance bounding. It has recently been noted that a
delayed-acceptance kernel need not be variance bounding even when its parent
is. We provide sufficient conditions for inheritance: for global algorithms,
such as the independence sampler, the error in the approximation should be
bounded; for local algorithms, two alternative sets of conditions are provided.
As a by-product of our initial, general result we also supply sufficient
conditions on any pair of proposals such that, for any shared target
distribution, if a Metropolis-Hastings kernel using one of the proposals is
variance bounding then so is the Metropolis-Hastings kernel using the other
proposal.