TY - JOUR

T1 - On the asymptotic efficiency of approximate Bayesian computation estimators

AU - Li, Wentao

AU - Fearnhead, Paul

N1 - This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Biometrika following peer review. The definitive publisher-authenticated version Wentao Li, Paul Fearnhead; On the asymptotic efficiency of approximate Bayesian computation estimators, Biometrika, Volume 105, Issue 2, 1 June 2018, Pages 285–299, https://doi.org/10.1093/biomet/asx078 is available online at: https://academic.oup.com/biomet/article/105/2/285/4818354

PY - 2018/6/1

Y1 - 2018/6/1

N2 - Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large-data limit. Our key assumption is that the data is summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation posterior. This result also shows that, in terms of asymptotic variance, we should use a summary statistic that is the same dimension as the parameter vector, p; and that any summary statistic of higher dimension can be reduced, through a lineartransformation, to dimension p in a way that can only reduce the asymptotic variance of the posterior mean. We look at how the Monte Carlo error of an importance sampling algorithm that samples from the approximate Bayesian computation posterior affects the accuracy of estimators. We give conditions on the importance sampling proposal distribution such that the variance of the estimator will be the same order as that of the maximum likelihood estimator based on the summary statistics used. This suggests an iterative importance sampling algorithm, which we evaluate empirically on a stochastic volatility model.

AB - Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large-data limit. Our key assumption is that the data is summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation posterior. This result also shows that, in terms of asymptotic variance, we should use a summary statistic that is the same dimension as the parameter vector, p; and that any summary statistic of higher dimension can be reduced, through a lineartransformation, to dimension p in a way that can only reduce the asymptotic variance of the posterior mean. We look at how the Monte Carlo error of an importance sampling algorithm that samples from the approximate Bayesian computation posterior affects the accuracy of estimators. We give conditions on the importance sampling proposal distribution such that the variance of the estimator will be the same order as that of the maximum likelihood estimator based on the summary statistics used. This suggests an iterative importance sampling algorithm, which we evaluate empirically on a stochastic volatility model.

KW - Approximate Bayesian computation

KW - Asymptotics

KW - Dimension reduction

KW - Importance Sampling

KW - Partial Information

KW - Proposal distribution

U2 - 10.1093/biomet/asx078

DO - 10.1093/biomet/asx078

M3 - Journal article

VL - 105

SP - 285

EP - 299

JO - Biometrika

JF - Biometrika

SN - 0006-3444

IS - 2

ER -