CONDUCTING A PROCESS TO ELICIT QUANTIFIED JUDGMENT

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So, the site characterization data reduces the odds from 1:4 to 1:12, and the corresponding probability from 0.2 to 0.076.

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Reliability modeling for assessing probabilities

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For some component events, engineering models are available for predicting behavior. In these cases, reliability analysis can be used to assess probabilities associated with the components. Reliability analysis propagates uncertainty in input parameters to uncertainties in predictions of performance. The assessment problem is changed from estimating probabilities of adverse performance directly to estimating probabilities for the input parameters. Once probabilities for the input parameters are assessed, any of a variety of simple mathematical techniques can be used to calculate probabilities associated with performance. Among these are rst-order second-moment approximations, advanced second-moment techniques, point-estimate calculations, or Monte Carlo simulation. Sometimes, experts elect to assess an additional component of uncertainty in the reliability analysis to account for model error. While there are many ways to do this, the most common is to assign a simple, unit-mean multiplier to the model output, having a standard deviation estimated by the experts to re ect model uncertainty. Experience with panels of experts suggests that model uncertainty is among the least tractable issues dealt with. The dif cult questions about model uncertainty have to do with underlying assumptions, with conceptualizations of physical processes, and with phenomenological issues. Experts tend to have strongly held beliefs on such matters, so that discussions can become intense. Nonetheless, model uncertainty is a critical aspect of risk assessment. Most models engineers deal with in their daily work were developed for design purposes. They deal with incipient failure conditions and with assuring that loads and resistances remain within working ranges. Risk assessment deals with adverse performance and failures. Thus, models which were developed to prevent the precursors of failure are now used to model failure processes themselves. Failure processes involve strongly nonlinear behaviors in considerations of time rates and sequences. Traditional engineering models may require a good deal of manipulation to draw conclusions about failure processes, and this typically requires a good deal of qualitative reasoning by experts.

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Correlations among uncertainties

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Most experts nd the notion of judgmental probability intuitively reasonable and, with practice, develop pro ciency at assessing their own uncertainty about individual events or parameters. On the other hand, most people, whether experts or not, have dif culty thinking about the correlations among the uncertainties pertaining to different events or parameters (viz. Alloy and Tabachnik 1984). Correlation means that a person s uncertainty about one event or parameter is affected by knowing whether another event occurred or by knowing the value of another parameter. People usually require signi cant analytical assistance when grappling with correlations, so it is good practice not to be overly aggressive in trying to assess correlations. The easiest way to assess probabilistic dependence between two uncertain quantities, x1 and x2 , is rst to assess the conditional probabilities for x2 , assuming various values for x1 , then

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EXPERT OPINION

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to assess the marginal probabilities for x1 (i.e. irrespective of the value of x2 ). The joint probabilities are found from the relationship P (x1 , x2 ) = P (x2 |x1 )P (x1 ) in which P (x1 , x2 ) = the probability of x1 and x2 occurring together, P (x2 |x1 ) = the probability of x2 , given the value of x1 , and P (x1 ) = the probability of x1 irrespective of the value of x2 . This approach requires that multiple assessments of the conditional probabilities P (x2 |x1 ) be made for various values of x1 , but the advantage is that the expert does not have to grapple explicitly with the concept of correlation coef cients, which tend not to be intuitive. The reverse conditional probabilities, P (x1 |x2 ), which are often needed for the risk assessment, can be calculated using Bayes Theorem. In practice, it is better to attempt to restructure a problem rather than to assess correlations among uncertainties. This can be done, for example, when two uncertainties are correlated because they each depend on some third uncertainty, as in the case of downstream costs of ooding. Emergency mobilization costs and property damage costs in the future may be correlated because each depends on in ation. It would be more effective to assess the uncertainties in each cost conditioned on in ation, and then to combine the two independent assessments, rather than to attempt to assess the correlated behavior of the two uncertainties. Of course, not all correlated uncertainties can be handled in this convenient way.

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