- davb_1_3_arxiv
**Rights statement:**The final publication is available at Springer via http://dx.doi.org/10.1007/s11009-021-09914-1Accepted author manuscript, 396 KB, PDF document

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- https://link.springer.com/article/10.1007/s11009-021-09914-1#article-info
Final published version

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

In: Methodology and Computing in Applied Probability, Vol. 24, No. 3, 30.09.2022, p. 2237-2260.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Sherlock, C & Lee, A 2022, 'Variance bounding of delayed-acceptance kernels', *Methodology and Computing in Applied Probability*, vol. 24, no. 3, pp. 2237-2260. https://doi.org/10.1007/s11009-021-09914-1

Sherlock, C., & Lee, A. (2022). Variance bounding of delayed-acceptance kernels. *Methodology and Computing in Applied Probability*, *24*(3), 2237-2260. https://doi.org/10.1007/s11009-021-09914-1

Sherlock C, Lee A. Variance bounding of delayed-acceptance kernels. Methodology and Computing in Applied Probability. 2022 Sept 30;24(3):2237-2260. Epub 2021 Nov 22. doi: 10.1007/s11009-021-09914-1

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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 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 $L^2$ 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 conditionson 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",

author = "Chris Sherlock and Anthony Lee",

note = "The final publication is available at Springer via http://dx.doi.org/10.1007/s11009-021-09914-1",

year = "2022",

month = sep,

day = "30",

doi = "10.1007/s11009-021-09914-1",

language = "English",

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journal = "Methodology and Computing in Applied Probability",

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AU - Sherlock, Chris

AU - Lee, Anthony

N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s11009-021-09914-1

PY - 2022/9/30

Y1 - 2022/9/30

N2 - 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 $L^2$ 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 conditionson 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.

AB - 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 $L^2$ 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 conditionson 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.

KW - Metropolis-Hastings

KW - Delayed-acceptance

KW - Variance bounding

KW - Conductance

U2 - 10.1007/s11009-021-09914-1

DO - 10.1007/s11009-021-09914-1

M3 - Journal article

VL - 24

SP - 2237

EP - 2260

JO - Methodology and Computing in Applied Probability

JF - Methodology and Computing in Applied Probability

SN - 1387-5841

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ER -