Risk Analysis, Vol. 21, No. 1, 2001 63 0272-4332/01/0200-0063$16.00/1 © 2001 Society for Risk Analysis Estimation of Error and Bias in Bayesian Monte Carlo Decision Analysis Using the Bootstrap Charles D. Linville, 1 Benjamin F. Hobbs, 2 * and Boddu N. Venkatesh 3 Bayesian Monte Carlo (BMC) decision analysis adopts a sampling procedure to estimate likelihoods and distributions of outcomes, and then uses that information to calculate the ex- pected performance of alternative strategies, the value of information, and the value of in- cluding uncertainty. These decision analysis outputs are therefore subject to sample error. The standard error of each estimate and its bias, if any, can be estimated by the bootstrap procedure. The bootstrap operates by resampling (with replacement) from the original BMC sample, and redoing the decision analysis. Repeating this procedure yields a distribution of decision analysis outputs. The bootstrap approach to estimating the effect of sample error upon BMC analysis is illustrated with a simple value-of-information calculation along with an analysis of a proposed control structure for Lake Erie. The examples show that the out- puts of BMC decision analysis can have high levels of sample error and bias. KEY WORDS: Bayesian Monte Carlo analysis; decision analysis; value of information; sample error; bootstrapping ing contaminated sediment (Dakins, Toll, Small, & Brand, 1996), greenhouse gas policy analysis (Linville, 1998), and water resources planning (Venkatesh & Hobbs, 1999). These applications included calculations of many of the typical outputs of decision analysis (Morgan & Henrion, 1990), including posterior proba- bilities, performance of alternatives, the expected value of perfect information (EVPI), the expected value of sample information (EVSI), the expected value of in- cluding uncertainty (EVIU), and quasi-option values. As an example, Dakins et al. (1996) have calcu- lated the EVSI for an environmental remediation de- cision for New Bedford (Massachusetts) Harbor by BMC analysis. Their application exemplifies the gen- eral BMC decision analysis procedure. They perform a Monte Carlo simulation of a decision model that cal- culates L(A|Y), the loss under alternative A and state of nature Y. Here, L(A|Y) includes dredging and dis- posal costs along with a penalty if the harbor requires additional remediation. In the BMC analysis, all un- 1 Department of Computer Science and Information Systems, American University, Washington, DC. 2 Department of Geography & Environmental Engineering, De- partment of Mathematical Sciences, The Johns Hopkins Univer- sity, Baltimore, MD. 3 ICF Consulting Group, Inc., Fairfax, VA. *Address correspondence to Benjamin F. Hobbs, Department of Geography & Environmental Engineering, Department of Math- ematical Sciences, The Johns Hopkins University, 3400 North Charles St., Baltimore, MD 21218; bhobbs@jhu.edu. 1. INTRODUCTION Bayesian Monte Carlo (BMC) analysis has proven to be a useful means of performing decision analyses for complex problems for which likelihood functions and distributions of benefits for alternatives cannot be obtained analytically. Instead, BMC uses Monte Carlo simulation to generate the required dis- tributions. Examples of application include updating of parameters in a sea level rise model (Patwardhan & Small, 1992), evaluation of alternatives for remediat-