Variance Bounding of Delayed-Acceptance Kernels

Chris Sherlock () and Anthony Lee
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Chris Sherlock: Lancaster University
Anthony Lee: University of Bristol

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 3, 2237-2260

Abstract: 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 associated “parent” Metropolis-Hastings kernel. Although the asymptotic variance of the ergodic average of any functional of the delayed-acceptance 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 the ergodic averages of all $$L^2$$ L 2 functionals of the chain are 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 non-local algorithms, such as the independence sampler, the discrepancy between the log density of the approximation and that of the truth 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.

Keywords: Metropolis-Hastings; Delayed-acceptance; Variance bounding; Conductance; 60J10; 65C40; 47A10 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-021-09914-1

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