We consider importance sampling (IS), pseudomarginal (PM), and delayed acceptance (DA) approaches to reversible Markov chain Monte Carlo, and compare the asymptotic variances. Despite their similarity in terms of using an approximation or unbiased estimators, not much has been said about the relative efficiency of IS and PM/DA. Simple examples demonstrate that the answer is setting specific. We show that the IS asymptotic variance is strictly less than a constant times the PM/DA asymptotic variance, where the constant is twice the essential supremum of the importance weight, and that the inequality becomes an equality as the weight approaches unity in the uniform norm. A version of the inequality also holds for the case of unbounded weight estimators, as long as the estimators are square integrable. The result, together with robustness and computational cost considerations in the context of parallel computing, lends weight to the suggestion of using IS over that of PM/DA when reasonabl 查看全文>>