Probability Bounds Analysis

Probability bounds analysis takes as inputs structures called p-boxes, which express sure bounds on a cumulative distribution function. One p-box is depicted in Figure 6.4. 

Probability bounds analysis combines p-boxes together in mathematical operations such as addition, subtraction, multiplication, and division. This is an alternative to what is usually done with Monte Carlo simulations, which usually evaluate a risk expression in one fell swoop in each iteration. In probability bounds analysis, a complex calculation is decomposed into its constituent arithmetic operations, which are computed separately to build up the final answer. The actual calculations needed to effect these operations with p-boxes are straightforward and elementary. This is not to say, however, that they are the kinds of calculations one would want to do by hand. In aggregate, they will often be cumbersome and should generally be done on computer. But it may be helpful to the reader to step through a numerical example just to see the nature of the calculation. 

In other cases, one may elect to use the best case scenario and make plans based on the left bound within the p-box. Which criterion one might use is outside the scope of probability bounds analysis. However, it should be emphasized that the possibilities within the bounds are not equivalent. And the analyst should not pick any answer from within these bounds. We recall the case of the engineers who designed Kansai International Airport on an island constructed with fill in a harbor near Kobe, Japan. They were reportedly told by geologists that the island would settle between 19 and 25 feet. They chose to plan for 19-foot subsidence, supposedly because planning for 25 feet would have been prohibitively expensive. One needs not be a student of Greek tragedy to anticipate the fate of such hubris. 

The KS limits are certainly a standard idea in probability theory and have been used in traditional risk analyses, for instance as a way to express the reliability of the results of a simulation. However, it has not heretofore been possible to use KS limits to characterize the statistical reliability of the inputs. There has been no way to propagate KS limits through calculations. Probability bounds analysis allows us to do this... 

There are 3 important limitations of probability bounds analysis. The 1st limitation is that, being only bounds on a distribution, a p-box cannot show what distribution is most likely within the box. A p-box provides no shades of gray or 2nd-order information that could tell us which distributions are the most probable. This is essentially the same problem, albeit at a higher level, that intervals had. It may, however, be possible to nest probability bounds analyses to get at the internal structure of the result. It is also often useful to simultaneously conduct a traditional Monte Carlo assessment, which will produce output distributions inside the output p-boxes. Together, these results characterize central and bounding estimates of the output distribution. 

Finally, although both probability bounds analysis and robust Bayes methods are fully legitimate applications of probability theory and, indeed, both find their foundations in classical results, they may be controversial in some quarters. Some argue that a single probability measure should be able to capture all of an individual s uncertainty. Walley (1991) has called this idea the dogma of ideal precision. The attitude has never been common in risk analysis, where practitioners are governed by practical considerations. However, the bounding approaches may precipitate some contention because they contradict certain attitudes about the universal applicability of pure probability. 

Model enveloping the outputs from alternative models can be combined using bounding methods (e.g., probability bounds analysis). 

Limitations on data availability are a recurrent concern in discussions about uncertainty analysis and probabilistic methods, but arguably these methods are most needed when data are limited. More work is required to develop tools, methods, and guidance for dealing with limited datasets. Specific aspects that require attention are the treatment of sampling error in probability bounds analysis, and the use of qualitative information and expert judgment. 

Table 6.3 lists the summary statistical measures yielded by 3 analyses of this hypothetical calculation. The 2nd column gives the results that might be obtained by a standard Monte Carlo analysis under an independence assumption (the dotted lines in Figure 6.7). The 3rd and 4th columns give results from probability bounding analyses, either with or without an assumption of independence. 

Recently, however, limited use of best estimate plus uncertainty analysis methods has been undertaken. This is consistent with the international trend toward use of such methods. In this approach, more physically realistic models, assumptions, and plant data are used to yield analysis predictions that are more representative of expected behavior. This requires a corresponding detailed analysis of the uncertainties in the analysis and their effect on the calculated consequences. Typically, the probability of meeting a specific numerical safety criterion, such as a fuel centerline temperature limit, is evaluated together with the confidence limit that results from the uncertainty distributions associated with governing analysis parameters. The best estimate plus uncertainties approach addresses many of the problematic issues associated with conservative bounding analysis by... 

In Figure 9 an analysis of the effect of allocating various amounts of resources to a business process is presented. This provides an assessment of both the probability bounds that a given resource level will lead to resource exhaustion, but also analysis of the non-functional property of the expected amount of time taken to reach such states. This type of provisioning analysis is frequently part of regulatory requirements for the introduction of new technologies (e.g. in the medical field (International Organization for Standardization 2003)) and the ability to perform this in an exact and automated fashion has been of considerable value to the project s industrial partner. 

Fortunately, the most severe weather is often very localized, so it is possible to examine the worst known storm near the reactor facility and use geometrical arguments to determine an estimate of the probability that the reactor site itself might be affected. Normally, a bounding analysis of that probability is sufficient to screen out most severe weather events from further consideration. The loss of offsite power as a result of severe weather is generally included in the overall loss of offsite power frequency (included in the internal events analysis). If any particular severe weather events can not be screened out based on low frequency, then analyses of plant response are performed during... 

A systematic difference is found, supported by indirect evidence that from experience precludes any explanation other than effect observed. This case does not necessarily call for a statistical evaluation, but an example will nonetheless be provided in the elemental analysis of organic chemicals (CHN analysis) reproducibilities of 0.2 to 0.3% are routine (for a mean of 38.4 wt-% C, for example, this gives a true value within the bounds 38.0. .. 38.8 wt-% for 95% probability). It is not out of the ordinary that traces of the solvent used in the... 

No sequence homologies can be detected. This is, perhaps, not surprising. The X-ray structure analysis of lysozyme by Phillips has shown that the polypeptide chain is folded in a way which puts none of the amino acids in sequential vicinity of the catalytic Asp-52 and Glu-37 that are near to the bound substrate. Comparable folding patterns can probably be realized with widely differing arrangements of amino acids, and thus the apparent lack of homologies. 

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