Computing Bayesian Means Using Simulation

This article is concerned with the estimation of α = E{r(Z)}, where Z is a random vector and the function values r(z) must be evaluated using simulation. Estimation problems of this form arise in the field of Bayesian simulation, where Z represents the uncertain (input) parameters of a system and r(z) is the expected performance of the system when Z = z. Our approach involves obtaining (possibly biased) simulation estimates of the function values r(z) for a number of different values of z, and then using a (possibly weighted) average of these estimates to estimate α. We start by considering the case where the chosen values of z are independent and identically distributed observations of the random vector Z (independent sampling). We analyze the resulting estimator as the total computational effort c grows and provide numerical results. Then we show that improved convergence rates can be obtained through the use of techniques other than independent sampling. Specifically, our results indicate that the use of quasi-random sequences yields a better convergence rate than independent sampling, and that in the presence of a suitable special structure, it may be possible to use other numerical integration techniques (such as Simpson’s rule) to achieve the best possible rate c^{− 1/2} as c → ∞. Finally, we present and analyze a general framework of estimators for α that encompasses independent sampling, quasi-random sequences, and Simpson’s rule as special cases.