Rights statement: © 2021 Institute of Mathematical Statistics
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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 - Efficiency of delayed-acceptance random walk Metropolis algorithms
AU - Sherlock, Chris
AU - Thiery, Alexandre
AU - Golightly, Andrew
N1 - © 2021 Institute of Mathematical Statistics
PY - 2021/10/15
Y1 - 2021/10/15
N2 - Delayed-acceptance Metropolis-Hastings and delayed-acceptance pseudo-marginal Metropolis-Hastings algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept-reject stage uses the cheap approximation for computing the Metropolis-Hastings ratio; proposals which are accepted at this stage are subjected to a further accept-reject step which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at thefirst stage, the cost of the algorithm is reduced and larger scalings may be used.We focus on the random walk Metropolis (RWM) and consider the delayed-acceptance RWM and the delayed-acceptance pseudo-marginal RWM. We provide a framework for incorporating relatively general deterministic approximations into the theoretical analysis of high-dimensional targets. Justified by diffusion-approximation arguments, we derive expressions for the limiting efficiency and acceptance rates in high dimensional settings. Finally, these theoretical insights are leveraged to formulate practical guidelines for the efficient tuning of the algorithms. The robustness of these guidelines and predictedproperties are verified against simulation studies, all of which are strictly outside of the domain of validity of our limit results.
AB - Delayed-acceptance Metropolis-Hastings and delayed-acceptance pseudo-marginal Metropolis-Hastings algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept-reject stage uses the cheap approximation for computing the Metropolis-Hastings ratio; proposals which are accepted at this stage are subjected to a further accept-reject step which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at thefirst stage, the cost of the algorithm is reduced and larger scalings may be used.We focus on the random walk Metropolis (RWM) and consider the delayed-acceptance RWM and the delayed-acceptance pseudo-marginal RWM. We provide a framework for incorporating relatively general deterministic approximations into the theoretical analysis of high-dimensional targets. Justified by diffusion-approximation arguments, we derive expressions for the limiting efficiency and acceptance rates in high dimensional settings. Finally, these theoretical insights are leveraged to formulate practical guidelines for the efficient tuning of the algorithms. The robustness of these guidelines and predictedproperties are verified against simulation studies, all of which are strictly outside of the domain of validity of our limit results.
U2 - 10.1214/21-AOS2068
DO - 10.1214/21-AOS2068
M3 - Journal article
VL - 49
SP - 2972
EP - 2990
JO - Annals of Statistics
JF - Annals of Statistics
SN - 0090-5364
IS - 5
ER -