Bayesian Monte Carlo Research Articles

Application of Bayesian Monte Carlo analysis to a Lagrangian photochemical air quality model

Uncertainties in ozone concentrations predicted with a Lagrangian photochemical air quality model have been estimated using Bayesian Monte Carlo (BMC) analysis. Bayesian Monte Carlo analysis provides a means of combining subjective “prior” uncertainty estimates developed by standard Monte Carlo techniques with information about the agreement between model outputs and observations. The resulting “posterior” uncertainty estimates reflect both the model's performance and subjective judgments about uncertainties in model parameters and inputs. To demonstrate the approach, BMC analysis was applied to a model of ozone concentrations along two-day trajectories ending on 28 August 1987 at Azusa and Riverside, CA. Refined estimates of uncertainties in base case O 3 concentrations were calculated, along with estimates of uncertainties in the response to 25% reductions in motor vehicle emissions of nitrogen oxides and volatile organic compounds. For the cases studied, the model results were in reasonable agreement with spatially interpolated observations. Bayesian updating reduced the estimated uncertainty in predicted peak O 3 concentrations from 35 to 20% at Azusa and from 24 to 18% at Riverside.

Analyzing investments for managing Lake Erie levels under climate change uncertainty

Analyses of investments that are irreversible and have uncertain benefits should consider the option of delaying a decision. For instance, the benefits of many water resource projects could change if global warming occurs. The magnitude of that warming is uncertain, and delaying projects until more information is available might be optimal. We examine whether this is true for construction of an outflow control structure for Lake Erie. Using Bayesian Monte Carlo (BMC)–based decision analysis, we find that considering climate uncertainty does make a difference. Climate change beliefs, in the form of prior distributions over transient climate scenarios, can affect the optimal strategy: in particular, climate change makes delaying construction more attractive. The option value of deferring the decision to build is as high as $20 million. Ignoring the possibility of climate warming can inflict an expected penalty as large as 20% of the cost of the control structure. We also compare climate risks to uncertainties in stage‐damage curves and find that they are approximately of equal importance.

Updating Uncertainty in an Integrated Risk Assessment: Conceptual Framework and Methods

Bayesian methods are presented for updating the uncertainty in the predictions of an integrated Environmental Health Risk Assessment (EHRA) model. The methods allow the estimation of posterior uncertainty distributions based on the observation of different model outputs along the chain of the linked assessment framework. Analytical equations are derived for the case of the multiplicative lognormal risk model where the sequential log outputs (log ambient concentration, log applied dose, log delivered dose, and log risk) are each normally distributed. Given observations of a log output made with a normally distributed measurement error, the posterior distributions of the log outputs remain normal, but with modified means and variances, and induced correlations between successive log outputs and log inputs. The analytical equations for forward and backward propagation of the updates are generally applicable to sums of normally distributed variables. The Bayesian Monte‐Carlo (BMC) procedure is presented to provide an approximate, but more broadly applicable method for numerically updating uncertainty with concurrent backward and forward propagation. Illustrative examples, presented for the multiplicative lognormal model, demonstrate agreement between the analytical and BMC methods, and show how uncertainty updates can propagate through a linked EHRA. The Bayesian updating methods facilitate the pooling of knowledge encoded in predictive models with that transmitted by research outcomes (e.g., field measurements), and thereby support the practice of iterative risk assessment and value of information appraisals.

Development of Bayesian Monte Carlo techniques for water quality model uncertainty

A new technique, Bayesian Monte Carlo (BMC), is used to quantify errors in water quality models caused by uncertain parameters. BMC also provides estimates of parameter uncertainty as a function of observed data on model state variables. The use of Bayesian inference generates uncertainty estimates that combine prior information on parameter uncertainty with observed variation in water quality data to provide an improved estimate of model parameter and output uncertainty. It also combines Monte Carlo analysis with Bayesian inference to determine the ability of random selected parameter sets to simulate observed data. BMC expands upon previous studies by providing a quantitative estimate of parameter acceptability using the statistical likelihood function. The likelihood of each parameter set is employed to generate an n-dimensional hypercube describing a probability distribution of each parameter and the covariance among parameters. These distributions are utilized to estimate uncertainty in model predictions. Application of BMC to a dissolved oxygen model reduced the estimated uncertainty in model output by 72% compared with standard Monte Carlo techniques. Sixty percent of this reduction was directly attributed to consideration of covariance between model parameters. A significant benefit of the technique is the ability to compare the reduction in total model output uncertainty corresponding to: (1) collection of more data on model state variables, and (2) laboratory or field studies to better define model processes. Limitations of the technique include computational requirements and accurate estimation of the joint probability distribution of model errors. This analysis was conducted assuming that model error is normally and independently distributed.