Bayesian Monte Carlo Research Articles

Bayesian Inference Based on Monte Carlo Technique for Multiplier of Performance Shaping Factor

Abstract The Human Error Probabilities (HEP) can be estimated using multipliers that correspond to the level of Performance Shaping Factors (PSFs) in the Human Reliability Analysis (HRA). This paper focuses on the adjustment of multipliers through Bayesian inference based on Monte Carlo techniques using the experimental results from simulators. Markov Chain Monte Carlo (MCMC) and Bayesian Monte Carlo (BMC) are used as Bayesian inference methods based on Monte Carlo techniques. MCMC is utilized to obtain the posterior distribution of the multipliers. BMC is used for the estimation of the moments of the posterior distribution such as the mean and variance. The results obtained by MCMC and that by BMC well agree with the reference results. As a case study, the data assimilation was performed using the results of the simulator experiment of Halden reactor. The results show that the multiplier changes by the result of a particular scenario and HEP of another scenario that uses the same multiplier also changes by data assimilation. Also, in the case study, the correlation between multipliers is obtained by the data assimilation and the correlation contributes to the reduction of uncertainty of HEP.

Two-Layer numerical model of soil moisture dynamics: Model assessment and Bayesian uncertainty estimation.

A two-layer model based on the integrated form of Richards' equation (RE) was recently developed to simulate the soil water movement in the roots layer and the vadose zone with a relatively shallow and dynamic water table. The model simulates thickness-averaged volumetric water content and matric suction as opposed to point values and was numerically verified for three soil textures using HYDRUS as a benchmark. However, the strengths and limitations of the two-layer model and its performance in stratified soils and under actual field conditions have not been tested. This study further examined the two-layer model using two numerical verification experiments and, most importantly, tested its performance at site-level under actual, highly variable hydroclimate conditions. Moreover, model parameters were estimated and uncertainty and sources of errors were quantified using a Bayesian framework. First, the two-layer model was evaluated for 231 soil textures under varying soil layer thicknesses with a uniform soil profile. Second, the two-layer model was assessed for stratified conditions where the top and bottom soil layers have contrasting hydraulic conductivities. The model was evaluated by comparing soil moisture and flux estimates to those from the HYDRUS model. Last, a case study of model application using data from a Soil Climate Analysis Network (SCAN) site was presented. Bayesian Monte Carlo (BMC) method was implemented for model calibration and quantifying sources of uncertainty under real hydroclimate and soil conditions. For a homogeneous soil profile, the two-layer model generally had excellent performance in estimating volumetric water content and fluxes, while the model performance slightly declined with increasing layer thickness and coarser textured soils. The model configurations regarding layer thicknesses and soil textures that generate accurate soil moisture and flux estimations were further suggested. With the two layers of contrasting permeability, model-simulated soil moisture contents and fluxes agreed well with those computed by HYDRUS, indicating that the two-layer model accurately handles the water flow dynamics around the layer interface. In the field application, given the highly variable hydroclimate conditions, the two-layer model combined with the BMC method showed good agreement with the observed average soil moisture of the root zone and the vadose zone below (RMSE <0.021 during calibration and <0.023 during validation periods). The contribution of parametric uncertainty to the total model uncertainty was too small compared to other sources. The numerical tests and the site level application showed that the two-layer model can reliably simulate thickness-averaged soil moisture and estimate fluxes in the vadose zone under various soil and hydroclimate conditions. Results also indicated that the BMC method could be a robust framework for vadose zone hydraulic parameters identification and model uncertainty estimation.

Bayesian inference of multi-group nuclear data by Monte Carlo sampling method

The multi-group nuclear data adjustment combines integral experiments with covariance data to reduce the crucial parameter uncertainties in the code verification and data validation. In this paper, the Monte Carlo sampling-based multi-group nuclear data adjustment methods, namely the unified Bayesian inference (UBI) method, the Unified Monte Carlo (UMC) method, and the Bayesian Monte Carlo (BMC) method, have been developed. These methods resort to statistical sampling to adjust the multi-group nuclear data and avoid sensitivity analysis. Besides, the sampling-based similarity analysis has been proposed to evaluate the neutronics similarity between experiments and industrial applications. Two simple pin-cells, regarded as an experiment and an industrial application respectively, are constructed to test the correctness and effectiveness of these methods. Besides, the B&W-1810 experiment with measurement values is chosen for the method verification. The results show that the posterior deviation and uncertainty see a dramatic decrease after the application of these adjustment methods.

Modeling the long-term persistence of neutralizing antibody in children and toddlers after vaccination with live attenuated Japanese encephalitis chimeric virus vaccine

ABSTRACTThe live-attenuated Japanese encephalitis chimeric virus vaccine JE-CV (IMOJEV®, Sanofi Pasteur) elicits a robust antibody response in children, which wanes over time. Clinical efficacy is based on a correlate of protection against JE infection defined as neutralizing antibody levels equal to or greater than the threshold of 10 (1/dil). Information on the duration of persistence of the JE antibody response above this threshold is needed. We constructed statistical models using 5-year persistence data from a randomised clinical trial (NCT00621764) in children (2–5 years old) primed with inactivated JE vaccine who received a booster dose of JE-CV, and in JE-naïve toddlers (12–24 months) who received a JE-CV single dose primary vaccination. Models were constructed using a Bayesian Monte-Carlo Markov Chain approach and implemented with OpenBugs V3.2.1. Antibody persistence was predicted for up to 10 years following JE-CV vaccination. Findings from a piecewise model with 2 phases (children) and a classic linear model (toddlers) are presented. For children, predicted median antibody titers (77 [2.5th–97.5th percentile range 41–144] 1/dil) remained above the threshold for seroprotection over the 10 years following booster JE-CV vaccination; the predicted median duration of protection was 19.5 years. For toddlers, 10 years after JE-CV primary vaccination median antibody titers were predicted to wane to around the level required for seroprotection (10.8 [5.8–20.1] 1/dil). A booster dose of JE-CV in children is predicted to provide long-term protection against JE. Such data are useful to facilitate decisions on implementation of and recommendations for future vaccination strategies.

Clinical high-resolution mapping of the proteoglycan-bound water fraction in articular cartilage of the human knee joint

PurposeWe applied our recently introduced Bayesian analytic method to achieve clinically-feasible in-vivo mapping of the proteoglycan water fraction (PgWF) of human knee cartilage with improved spatial resolution and stability as compared to existing methods. Materials and methodsMulticomponent driven equilibrium single-pulse observation of T1 and T2 (mcDESPOT) datasets were acquired from the knees of two healthy young subjects and one older subject with previous knee injury. Each dataset was processed using Bayesian Monte Carlo (BMC) analysis incorporating a two-component tissue model. We assessed the performance and reproducibility of BMC and of the conventional analysis of stochastic region contraction (SRC) in the estimation of PgWF. Stability of the BMC analysis of PgWF was tested by comparing independent high-resolution (HR) datasets from each of the two young subjects. ResultsUnlike SRC, the BMC-derived maps from the two HR datasets were essentially identical. Furthermore, SRC maps showed substantial random variation in estimated PgWF, and mean values that differed from those obtained using BMC. In addition, PgWF maps derived from conventional low-resolution (LR) datasets exhibited partial volume and magnetic susceptibility effects. These artifacts were absent in HR PgWF images. Finally, our analysis showed regional variation in PgWF estimates, and substantially higher values in the younger subjects as compared to the older subject. ConclusionsBMC-mcDESPOT permits HR in-vivo mapping of PgWF in human knee cartilage in a clinically-feasible acquisition time. HR mapping reduces the impact of partial volume and magnetic susceptibility artifacts compared to LR mapping. Finally, BMC-mcDESPOT demonstrated excellent reproducibility in the determination of PgWF.

On the use of Bayesian Monte-Carlo in evaluation of nuclear data

As model parameters, necessary ingredients of theoretical models, are not always predicted by theory, a formal mathematical framework associated to the evaluation work is needed to obtain the best set of parameters (resonance parameters, optical models, fission barrier, average width, multigroup cross sections) with Bayesian statistical inference by comparing theory to experiment. The formal rule related to this methodology is to estimate the posterior density probability function of a set of parameters by solving an equation of the following type: pdf(posterior) ∼ pdf(prior) × a likelihood function. A fitting procedure can be seen as an estimation of the posterior density probability of a set of parameters (referred as x→ ) knowing a prior information on these parameters and a likelihood which gives the probability density function of observing a data set knowing x→ . To solve this problem, two major paths could be taken: add approximations and hypothesis and obtain an equation to be solved numerically (minimum of a cost function or Generalized least Square method, referred as GLS) or use Monte-Carlo sampling of all prior distributions and estimate the final posterior distribution. Monte Carlo methods are natural solution for Bayesian inference problems. They avoid approximations (existing in traditional adjustment procedure based on chi-square minimization) and propose alternative in the choice of probability density distribution for priors and likelihoods. This paper will propose the use of what we are calling Bayesian Monte Carlo (referred as BMC in the rest of the manuscript) in the whole energy range from thermal, resonance and continuum range for all nuclear reaction models at these energies. Algorithms will be presented based on Monte-Carlo sampling and Markov chain. The objectives of BMC are to propose a reference calculation for validating the GLS calculations and approximations, to test probability density distributions effects and to provide the framework of finding global minimum if several local minimums exist. Application to resolved resonance, unresolved resonance and continuum evaluation as well as multigroup cross section data assimilation will be presented.

Rapid simultaneous high-resolution mapping of myelin water fraction and relaxation times in human brain using BMC-mcDESPOT

A number of central nervous system (CNS) diseases exhibit changes in myelin content and magnetic resonance longitudinal, T1, and transverse, T2, relaxation times, which therefore represent important biomarkers of CNS pathology. Among the methods applied for measurement of myelin water fraction (MWF) and relaxation times, the multicomponent driven equilibrium single pulse observation of T1 and T2 (mcDESPOT) approach is of particular interest. mcDESPOT permits whole brain mapping of multicomponent T1 and T2, with data acquisition accomplished within a clinically realistic acquisition time. Unfortunately, previous studies have indicated the limited performance of mcDESPOT in the setting of the modest signal-to-noise range of high-resolution mapping, required for the depiction of small structures and to reduce partial volume effects. Recently, we showed that a new Bayesian Monte Carlo (BMC) analysis substantially improved determination of MWF from mcDESPOT imaging data. However, our previous study was limited in that it did not discuss determination of relaxation times. Here, we extend the BMC analysis to the simultaneous determination of whole-brain MWF and relaxation times using the two-component mcDESPOT signal model. Simulation analyses and in-vivo human brain studies indicate the overall greater performance of this approach compared to the stochastic region contraction (SRC) algorithm, conventionally used to derive parameter estimates from mcDESPOT data. SRC estimates of the transverse relaxation time of the long T2 fraction, T2,l, and the longitudinal relaxation time of the short T1 fraction, T1,s, clustered towards the lower and upper parameter search space limits, respectively, indicating failure of the fitting procedure. We demonstrate that this effect is absent in the BMC analysis. Our results also showed improved parameter estimation for BMC as compared to SRC for high-resolution mapping. Overall we find that the combination of BMC analysis and mcDESPOT, BMC-mcDESPOT, shows excellent performance for accurate high-resolution whole-brain mapping of MWF and bi-component transverse and longitudinal relaxation times within a clinically realistic acquisition time.

A Bayesian Monte Carlo-based algorithm for the estimation of small failure probabilities of systems affected by uncertainties

The estimation of system failure probabilities in presence of uncertainties may be a difficult task when the values involved are very small, so that sampling-based Monte Carlo methods may become computationally impractical, especially if the computer codes used to model the system response require large computational efforts, both in terms of time and memory. In this work, we propose to exploit the Bayesian Monte Carlo (BMC) approach to the estimation of definite integrals for developing a new, efficient algorithm for estimating small failure probabilities. The Bayesian framework allows an effective use of all the information available, i.e. the computer code evaluations and the input uncertainty distributions, and, at the same time, the analytical formulation of the Bayesian estimator guarantees the construction of a computationally lean algorithm. The proposed method is first satisfactorily tested with reference to an analytic, two-dimensional case study of literature, offering satisfactory results; then, it is applied to a realistic case study of a natural convection-based cooling system of a gas-cooled fast reactor, operating under a post-loss-of-coolant accident (LOCA), showing performances comparable to those of other efficient alternative methods of literature.

A Comparison of Bayesian Methods for Uncertainty Analysis in Hydraulic and Hydrodynamic Modeling

AbstractWe evaluate and compare the performance of Bayesian Monte Carlo (BMC), Markov chain Monte Carlo (MCMC), and the Generalized Likelihood Uncertainty Estimation (GLUE) for uncertainty analysis in hydraulic and hydrodynamic modeling (HHM) studies. The methods are evaluated in a synthetic 1D wave routing exercise based on the diffusion wave model, and in a multidimensional hydrodynamic study based on the Environmental Fluid Dynamics Code to simulate estuarine circulation processes in Weeks Bay, Alabama. Results show that BMC and MCMC provide similar estimates of uncertainty. The posterior parameter densities computed by both methods are highly consistent, as well as the calibrated parameter estimates and uncertainty bounds. Although some studies suggest that MCMC is more efficient than BMC, our results did not show a clear difference between the performance of the two methods. This seems to be due to the low number of model parameters typically involved in HHM studies, and the use of the same likelihood function. In fact, for these studies, the implementation of BMC results simpler and provides similar results to MCMC. The results of GLUE are, on the other hand, less consistent to the results of BMC and MCMC in both applications. The posterior probability densities tend to be flat and similar to the uniform priors, which can result in calibrated parameter estimates centered in the parametric space.

An open source Bayesian Monte Carlo isotope mixing model with applications in Earth surface processes

AbstractThe implementation of isotopic tracers as constraints on source contributions has become increasingly relevant to understanding Earth surface processes. Interpretation of these isotopic tracers has become more accessible with the development of Bayesian Monte Carlo (BMC) mixing models, which allow uncertainty in mixing end‐members and provide methodology for systems with multicomponent mixing. This study presents an open source multiple isotope BMC mixing model that is applicable to Earth surface environments with sources exhibiting distinct end‐member isotopic signatures. Our model is first applied to new δ18O and δD measurements from the Athabasca Glacier, which showed expected seasonal melt evolution trends and vigorously assessed the statistical relevance of the resulting fraction estimations. To highlight the broad applicability of our model to a variety of Earth surface environments and relevant isotopic systems, we expand our model to two additional case studies: deriving melt sources from δ18O, δD, and 222Rn measurements of Greenland Ice Sheet bulk water samples and assessing nutrient sources from ɛNd and 87Sr/86Sr measurements of Hawaiian soil cores. The model produces results for the Greenland Ice Sheet and Hawaiian soil data sets that are consistent with the originally published fractional contribution estimates. The advantage of this method is that it quantifies the error induced by variability in the end‐member compositions, unrealized by the models previously applied to the above case studies. Results from all three case studies demonstrate the broad applicability of this statistical BMC isotopic mixing model for estimating source contribution fractions in a variety of Earth surface systems.

When will the TBT go away? Integrating monitoring and modelling to address TBT's delayed disappearance in the Drammensfjord, Norway

Despite a substantial decrease in the use and production of the marine antifouling agent tributyltin (TBT), its continuing presence in harbors remains a serious environmental concern. Herein a case study of TBT's persistence in the Drammensfjord, Norway, is presented. In 2005, severe TBT pollution was measured in the harbor of the Drammensfjord, with an average sediment concentration of 3387 μg kg−1. To chart natural recovery in the Drammensfjord, an extensive sampling campaign was carried out over six years (2008–2013), quantifying TBT in water, settling particles and sediments. The monitoring campaign found a rapid decrease in sediment TBT concentration in the most contaminated areas, as well as a decrease in TBT entering the harbor via rivers and urban runoff. Changes observed in the more remote areas of the Drammensfjord, however, were less substantial. These data, along with measured and estimated geophysical properties, were used to parameterize and calibrate a coupled linear water-sediment model, referred to as the Drammensfjord model, to make prognosis on future TBT levels due to natural recovery. Unique to this type of model, the calibration was done using a Bayesian Monte Carlo (BMC) updating approach, which used monitoring data to calibrate predictions, as well as reduce the uncertainty of input parameters. To our knowledge, this is the first use of BMC updating to calibrate a model describing natural recovery in a lake/harbor type system. Prior to BMC updating, the non-calibrated model data agreed with monitoring data by a factor of 4.3. After BMC updating, the agreement was within a factor 3.2. The non-calibrated model predicted an average sediment concentration in the year 2025 of 2.5 μg kg−1. The BMC calibrated model, however, predicted a higher concentration in the year 2025 of 16 μg kg−1. This discrepancy was mainly due to the BMC calibration increasing the estimated riverine and runoff TBT emission levels relative to the initial input levels. Future monitoring campaigns can be used for further calibration of emission levels, and a clearer prognosis of when natural recovery will remove TBT pollution.

Bayesian Monte Carlo for evaluation of uncertainty in hydrodynamic models of coastal systems

Camacho, R.A. and Martin, J.L., 2013. Bayesian Monte Carlo for evaluation of uncertainty in hydrodynamic models of coastal systems. In: Conley, D.C., Masselink, G., Russell, P.E. and O’Hare, T.J. (eds.), Proceedings 12 th International Coastal Symposium (Plymouth, England), Journal of Coastal Research, Special Issue No. 65, pp. 886-891, ISSN 07490208. Uncertainty analysis constitutes the set of procedures and strategies conducted to identify, quantify, and report the impacts of different sources of errors on the predictions of a numerical model. Although during the last couple of decades several investigations in different areas of water resources have explicitly discussed the importance of a new modeling paradigm where model predictions are reported along with uncertainty estimates, uncertainty analysis remains an emerging topic in the field of hydrodynamic modeling. Presently, several unresolved issues remain with regard to the applicability, benefits, and limitations of existing strategies for quantification of uncertainty, and the identification of the principal sources of uncertainty in practical applications. In this document we apply the Bayesian Monte Carlo (BMC) method as a strategy to perform uncertainty analysis in coastal modeling studies. BMC is based on robust principles of Bayesian inference and Monte Carlo simulations, and has been successfully applied for studies of groundwater modeling, hydrologic modeling, and water quality modeling in the past. The procedural aspects of the method are discussed in detail, and the implementation of the method is demonstrated for the hydrodynamic model of the St. Louis Bay estuary, Mississippi (USA) for the evaluation of the impacts of input data errors. Results indicate that BMC is an effective strategy for uncertainty analyses, although in professional practice its use may be limited by computational requirements. We also point out that the effectiveness of the method may be affected by the correct selection of the likelihood function and the model error variance.

Joint inversion of surface wave dispersion and receiver functions: a Bayesian Monte-Carlo approach

A non-linear Bayesian Monte-Carlo method is presented to estimate a Vsv model beneath stations by jointly interpreting Rayleigh wave dispersion and receiver functions and associated uncertainties. The method is designed for automated application to large arrays of broad-band seismometers. As a testbed for the method, 185 stations from the USArray Transportable Array are used in the IntermountainWest, a region that is geologically diverse and structurally complex. Ambient noise and earthquake tomography are updated by applying eikonal and Helmholtz tomography, respectively, to construct Rayleighwave dispersion maps from 8 to 80 s across the study region with attendant uncertainty estimates.Amethod referred to as ‘harmonic stripping method’ is described and applied as a basis for quality control and to generate backazimuth independent receiver functions for a horizontally layered, isotropic effective medium with uncertainty estimates for each station. A smooth parametrization between (as well as above and below) discontinuities at the base of the sediments and crust suffices to fit most features of both data types jointly across most of the study region. The effect of introducing receiver functions to surface wave dispersion data is quantified through improvements in the posterior marginal distribution of model variables. Assimilation of receiver functions quantitatively improves the accuracy of estimates of Moho depth, improves the determination of the Vsv contrast across Moho, and improves uppermost mantle structure because of the ability to relax a priori constraints. The method presented here is robust and can be applied systematically to construct a 3-D model of the crust and uppermost mantle across the large networks of seismometers that are developing globally, but also provides a framework for further refinements in the method.

Safety assessment of infrastructures using a new Bayesian Monte Carlo method

A recently developed Bayesian Monte Carlo (BMC) method and its application to safety assessment of structures are described in this paper. We use a one-dimensional BMC method that was proposed in 2009 by Rajabalinejad in order to develop a weighted logical dependence between successive Monte Carlo simulations. Our main objective in this research is to show that the extended BMC can dramatically improve simulation efficiency by using prior information from modelling and outcomes of preceding simulations. We provide theory and numerical algorithms for an extended BMC method for multi-dimensional problems, integrate it with a probabilistic finite element model and apply these coupled models to assessment of reliability of a flood defence for the 17th Street Flood Wall system in New Orleans. This is the first successful demonstration of the BMC method to a complex system. We provide a comparison of the numerical efficiency for the BMC, Monte Carlo (MC) and Dynamic Bounds methods that are used in reliability assessment of complex infrastructures.

Prognostics of lithium-ion batteries based on Dempster–Shafer theory and the Bayesian Monte Carlo method

A new method for state of health (SOH) and remaining useful life (RUL) estimations for lithium-ion batteries using Dempster–Shafer theory (DST) and the Bayesian Monte Carlo (BMC) method is proposed. In this work, an empirical model based on the physical degradation behavior of lithium-ion batteries is developed. Model parameters are initialized by combining sets of training data based on DST. BMC is then used to update the model parameters and predict the RUL based on available data through battery capacity monitoring. As more data become available, the accuracy of the model in predicting RUL improves. Two case studies demonstrating this approach are presented.

Bayesian Monte Carlo method

To reduce cost of Monte Carlo (MC) simulations for time-consuming processes, Bayesian Monte Carlo (BMC) is introduced in this paper. The BMC method reduces number of realizations in MC according to the desired accuracy level. BMC also provides a possibility of considering more priors. In other words, different priors can be integrated into one model by using BMC to further reduce cost of simulations. This study suggests speeding up the simulation process by considering the logical dependence of neighboring points as prior information. This information is used in the BMC method to produce a predictive tool through the simulation process. The general methodology and algorithm of BMC method are presented in this paper. The BMC method is applied to the simplified break water model as well as the finite element model of 17th Street Canal in New Orleans, and the results are compared with the MC and Dynamic Bounds methods.

Influence of indoor transport and mixing time scales on the performance of sensor systems for characterizing contaminant releases

Optimizing real-time sensor systems to detect and identify relevant characteristics of an indoor contaminant event is a challenging task. The interpretation of incoming sensor data is confounded by uncertainties in building operation, in the forces driving contaminant transport, and in the physical parameters governing transport. In addition, simulation tools used by the sensor interpretation algorithm introduce modeling uncertainties. This paper explores how the time scales inherent in contaminant transport influence the information that can be extracted from real-time sensor data. In particular, we identify three time scales (within room mixing, room-to-room transport, and removal from the building) and study how they affect the ability of a Bayesian Monte Carlo (BMC) sensor interpretation algorithm to identify the release location and release mass from a set of experimental data, recorded in a multi-floor building. The research shows that some limitations in the BMC approach do not depend on details of the models or the algorithmic implementation, but rather on the physics of contaminant transport. These inherent constraints have implications for the design of sensor systems.

On Monte Carlo methods for Bayesian inference

Bayesian methods are experiencing increased use for probabilistic ecological modelling. Most Bayesian inference requires the numerical approximation of analytically intractable integrals. Two methods based on Monte Carlo simulation have appeared in the ecological/environmental modelling literature. Though they sound similar, the Bayesian Monte Carlo (BMC) and Markov Chain Monte Carlo (MCMC) methods are very different in their efficiency and effectiveness in providing useful approximations for accurate inference in Bayesian applications. We compare these two methods using a low-dimensional biochemical oxygen demand decay model as an example. We demonstrate that the BMC is extremely inefficient because the prior parameter distribution, from which the Monte Carlo sample is drawn, is often a poor surrogate for the posterior parameter distribution, particularly if the parameters are highly correlated. In contrast, MCMC generates a chain that converges, in distribution, on the posterior parameter distribution, that can be regarded as a sample from the posterior distribution. The inefficiency of the BMC can lead to marginal posterior parameter distributions that appear irregular and may be highly misleading because the important region of the posterior distribution may never be sampled. We also point out that a priori specification of the model error variance can strongly influence the estimation of the principal model parameters. Although the BMC does not require that the model error variance be specified, most published applications have treated this variance as a known constant. Finally, we note that most published BMC applications have chosen a uniform prior distribution, making the BMC more similar to a likelihood-based inference rather than a Bayesian method because the posterior is unaffected by the prior. Though other prior distributions could be applied, the treatment of Monte Carlo samples with any other choice of prior distribution has not been discussed in the BMC literature.

Estimation of error and bias in Bayesian Monte Carlo decision analysis using the bootstrap.

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 expected performance of alternative strategies, the value of information, and the value of including 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 outputs of BMC decision analysis can have high levels of sample error and bias.

Representing uncertainty in climate change scenarios: a Monte-Carlo approach

Climate change impact assessment is subject to a range of uncertainties due to both incomplete and unknowable knowledge. This paper presents an approach to quantifying some of these uncertainties within a probabilistic framework. A hierarchical impact model is developed that addresses uncertainty about future greenhouse gas emissions, the climate sensitivity, and limitations and unpredictability in general circulation models. The hierarchical model is used in Bayesian Monte-Carlo simulations to define posterior probability distributions for changes in seasonal-mean temperature and precipitation over the United Kingdom that are conditional on prior distributions for the model parameters. The application of this approach to an impact model is demonstrated using a hydrological example.