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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Variance bounding of delayed-acceptance kernels
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
IS - 3
ER -