Probabilistic networks are now fairly well established as practical representations of knowledge for reasoning under uncertainty, as demonstrated by an increasing number of successful applications in such domains as (medical) diagnosis and prognosis, planning, vision,

When building a Bayesian belief network, usually a large number of probabilities have to be assessed by experts in the domain of application. Experience shows that experts often are reluctant to assess all probabilities required, feeling that they are unable to give assessments with a high level of accuracy. We argue that the elicitation of probabilities from experts can be supported to a large extent by iteratively performing sensitivity analyses of the belief network in the making, starting with rough, initial assessments. Since it gives insight into which probabilities require a high level of accuracy and which do not, performing a sensitivity analysis allows for focusing further elicitation efforts. We propose an elicitation procedure in which, alternatingly, sensitivity analyses are performed and probability assessments are refined, until satisfactory behaviour of the belief network is obtained, until the costs of further elicitation outweigh the benefits of higher accur...

Elicitation is a key task for subjectivist Bayesians. While skeptics hold that it cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subjectmatter-expert colleagues. This paper reviews the state-of-the-art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively, and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful, i.e. what criteria should be employed? Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily “true ” in some objectivistic sense, and cannot be judged that way. We see elicitation as simply part of the process of statistical modeling. Indeed in a hierarchical model it is ambiguous at which point the likelihood ends and the prior begins. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions.

Most quantitative studies of climate-change policy attempt to predict the greenhouse-gas reduction plan that will have the optimum balance of longterm costs and benefits. We find that the large uncertainties associated with the climate-change problem can make the policy prescriptions of this traditional approach unreliable. In this study, we construct a large uncertainty space that includes the possibility of large and/or abrupt climate changes and/or of technology breakthroughs that radically reduce projected abatement costs. We use computational experiments on a linked system of climate and economic models to compare the performance of a simple adaptive strategy --- one that can make midcourse corrections based on observations of the climate and economic systems --- and two commonly advocated best-estimate policies based ondifferent expectations about the long-term consequences of climate change. We find that the Do-a-Little and Emissions-Stabilization best-estimate policies perform well in therespective regions of the uncertainty space where their estimates are valid, but can fail severely in those regions where their estimates are wrong. In contrast, the adaptive strategy can make midcourse corrections and avoid significant errors. While its success is no surprise, the adaptive-strategy approach provides an analytic framework to examine important policy and research issues that will likely arise as society adapts to climate change, and which cannot be easily addressed in studies using best-estimate approaches. 1.

This report addresses the characterization of measurements that include epistemic uncertainties in the form of intervals. It reviews the application of basic descriptive statistics to data sets which contain intervals rather than exclusively point estimates. It describes algorithms to compute various means, the median and other percentiles, variance, interquartile range, moments, confidence limits, and other important statistics and summarizes the computability of these statistics as a function of sample size and characteristics of the intervals in the data (degree of overlap, size and regularity of widths, etc.). It also reviews the prospects for analyzing such data sets with the methods of inferential statistics such as outlier detection and regressions. The report explores the tradeoff between measurement precision and sample size in statistical results that are sensitive to both. It also argues that an approach based on interval statistics could be a reasonable alternative to current standard methods for evaluating, expressing and propagating measurement uncertainties.

[1] This paper examines the health crisis in Bangladesh due to dissolved arsenic in groundwater. First, we use geostatistical methods to construct a map of arsenic concentrations that divides Bangladesh into regions and estimate vertical concentration trends in these regions. Then, we use census data to estimate exposure distributions in the regions; we use epidemiological data from West Bengal and Taiwan to estimate dose response functions for arsenicosis and arsenic-induced cancers; and we combine the regional exposure distributions and the dose response models to estimate the health effects of groundwater arsenic in Bangladesh. We predict that long-term exposure to present arsenic concentrations will result in approximately 1,200,000 cases of hyperpigmentation, 600,000 cases of keratosis, 125,000 cases of skin cancer, and 3000 fatalities per year from internal cancers. Although these estimates are very uncertain, the method provides a framework for incorporating better data as it becomes available. Moreover, we examine the remedy of drilling deeper wells in selected regions of Bangladesh. By replacing 31% of the wells in the country with deeper wells the health effects of drinking groundwater arsenic could be reduced by approximately 70 % provided that arsenic concentrations in

We develop a method for assessing uncertainty about quantities of interest using urban simulation models. The method is called Bayesian melding, and extends a previous method developed for macrolevel deterministic simulation models to agent-based stochastic models. It encodes all the available information about model inputs and outputs in terms of prior probability distributions and likelihoods, and uses Bayes’s theorem to obtain the resulting posterior distribution of any quantity of interest that is a function of model inputs and/or outputs. It is Monte Carlo based, and quite easy to implement. We applied it to the projection of future household numbers by traffic activity zone in Eugene-Springfield, Oregon, using the UrbanSim model developed at the University of Washington. We compared it with a simpler method that uses repeated runs of the model with fixed estimated inputs. We found that the simple repeated runs method gave distributions of quantities of interest that were too narrow, while Bayesian melding gave well calibrated uncertainty statements.

We consider a panel of experts asked to assign probabilities to events, both logically simple and complex. The events evaluated by different experts are based on overlapping sets of variables but may otherwise be distinct. The union of all the judgments will likely be probabilistic incoherent. We address the problem of revising the probability estimates of the panel so as to produce a coherent set that best represents the group's expertise.

The average probability estimate of J> 1 judges is generally better than its components. Two studies test 3 predictions regarding averaging that follow from theorems based on a cognitive model of the judges and idealizations of the judgment situation. Prediction 1 is that the average of conditionally pairwise independent estimates will be highly diagnostic, and Prediction 2 is that the average of dependent estimates (differing only by independent error terms) may be well calibrated. Prediction 3 contrasts between- and within-subject averaging. Results demonstrate the predictions ' robustness by showing the extent to which they hold as the information conditions depart from the ideal and as J increases. Practical consequences are that (a) substantial improvement can be obtained with as few as 2- 6 judges and (b) the decision maker can estimate the nature of the expected improvement by considering the information conditions. On many occasions, experts are required to provide decision makers or policymakers with subjective probability estimates of uncertain events (Morgan & Henrion, 1990). The extensive literature (e.g., Harvey, 1997; McClelland & Bolger, 1994) on the topic shows that in general, but with clear exceptions, subjective

Sensitivity analysis is a method to investigate the effects of varying a model's parameters on its predictions. It was recently suggested as a suitable means to facilitate quantifying the joint probability distribution of a Bayesian belief network. This article presents practical experience with performing sensitivity analyses on a belief network in the field of medical prognosis and treatment planning. Three network quantifications with different levels of informedness were constructed. Two poorly-informed quantifications were improved by replacing the most influential parameters with the corresponding parameter estimates from the well-informed network quantification; these influential parameters were found by performing one-way sensitivity analyses. Subsequently, the results of the replacements were investigated by comparing network predictions. It was found that it may be sufficient to gather a limited number of highly-informed network parameters to obtain a satisfying network quant...