This tutorial introduces the notions of interval-valued probability and imprecisely specified probability distributions and reviews their uses in risk analysis. It will address the approaches of interval probabilities, probability bounds analysis, Dempster-Shafer theory, robust Bayes methods, and the theory of imprecise probabilities. It will also illustrate how sparse, incomplete or imprecise data can be used to fashion inputs for these models.
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This full-day tutorial introduces the notions of interval-valued probabilities and imprecisely specified probability distributions and their uses in risk analysis. It reviews five practical and quantitative approaches based on these elementary notions. The simplest approach uses the idea of interval probability, in which the probability of an event can be specified as an interval of possible values rather than only as a precise one. This idea, dating from George Boole, provides a convenient way to assess the reliability of fault-tree risk analyses. This idea is generalized by probability bounds analysis, which propagates constraints on a distribution function through mathematical operations, and Dempster-Shafer theory which recognizes that uncertainty attending any real-world measurement may not allow an analyst to distinguish between events in empirical evidence. These approaches are related to robust Bayes (aka Bayesian sensitivity) methods, in which an analyst can relax the requirement that the prior distribution and likelihood function must be precisely specified. The most general approach comes from the theory of imprecise probabilities in which uncertainty is represented by closed, convex sets of probability distributions.
These five approaches redress, or comprehensively solve, several major deficiencies of Monte Carlo simulations and of standard probability theory in risk assessments. For instance, it is almost always difficult, if not impossible, to completely characterize dependencies among variables in a risk analysis. As a result, in the practical situations where empirical data are limiting, these difficulties can result in assessments that are arbitrarily over-specified and therefore misleading. More fundamentally, it can be argued that probability theory has an inadequate model of ignorance because it uses equiprobability as a model for incertitude and thus cannot distinguish uniform risk from pure lack of knowledge. In most practical risk assessments, some uncertainty is epistemic rather than aleatory, that is, it is incertitude rather than variability. For example, uncertainty about the shape of a probability distribution and most other instances of model uncertainty are typically epistemic. Treating incertitude as though it were variability is even worse than overspecification because it confounds epistemic and aleatory uncertainty and leads to risk conclusions that are simply wrong. Approaches based on imprecise probabilities allow an analyst to keep these kinds of uncertainty separate and treat them differently as necessary to maintain the interpretation of risk as the frequency of adverse outcomes.
The interval and imprecise probability methods also make backcalculations possible and practicable. Backcalculation is required to compute cleanup goals, remediation targets and performance standards from available knowledge and constraints about uncertain variables. The needed calculations are notoriously difficult with standard probabilistic methods and cannot be done at all with straightforward Monte Carlo simulation.
Although the five approaches arose from distinct scholarly traditions and have many important differences, the tutorial emphasizes that they share a commonality of purpose and employ many of the same ideas and methods. They can be viewed as complementary, and they constitute a single perspective on risk analysis that is sharply different from both traditional worst-case and standard probabilistic approaches. Each approach is illustrated with a numerical case study and summarized by a checklist of reasons to use, and not to use, the approach.
The workshop will also review how analysis of data sets can benefit from adopting imprecise probability methods. The task, sometimes called "knowledge discovery", of inferring models that make knowledge about a subject explicit typically assumes that empirical data are the only source of information. Thus, learning from data is presumed to start under a condition of prior ignorance. The available data are often imprecise due to measurement uncertainty or incomplete, such as when values are missing in the data set. This constitutes another form of ignorance that is about the data themselves. Modeling these forms of ignorance carefully is a central feature needed to make models and their applications reliable. The workshop will show how imprecise probability allows one to reliably handle incomplete data in a way that significantly departs from established approaches. This depends on the ability to work with weak assumptions about the data.
The presentation style will be casual and interactive. Participants will receive a CD of the illustrations used during the tutorial.
Scott Ferson is Senior Scientist at Applied Biomathematics. His research focuses on developing reliable mathematical and statistical tools for risk assessments and on methods for uncertainty analysis when empirical information is very sparse. Ferson holds a Ph.D. from the State University of New York at Stony Brook. He is author of RAMAS Risk Calc Software 4.0: Risk Assessment with Uncertain Numbers (Lewis Publishers). He has written over 75 other scholarly publications, including four other books and several software packages, in environmental risk analysis and uncertainty propagation. His research has addressed quality assurance for Monte Carlo assessments, exact methods for detecting clusters in small data sets, backcalculation methods for use in remediation planning, and distribution-free methods of risk analysis appropriate for use in information-poor situations.
The registration fee is $175 before 10 November, or $205 on site. You do not need to register for the Annual Meeting to attend the workshop. Registration will be handled by
The event will be held 9:00am - 5:30pm on Sunday, 5 December 2004, at
The room for the event has not yet been determined; check with the hotel concierge. To reserve a room at the hotel, call 760-322-6000 before 4 November 2004. Be sure to identify yourself as a SRA Annual Meeting attendee to receive the SRA group rate of $129 per night (single or double occupancy) plus 13.55% tax. There is also an $8 per day resort fee which includes parking and local calls and an 89 cent Palm Springs utility tax. Cancellations must be made at least 72 hours in advance. See a description of the hotel at http://www.wyndham.com/hotels/PSPPS/main.wnt.
More information can be obtained from Scott Ferson scott@ramas.com, telephone 631-751-4350, fax 631-751–3435.
The Imprecise Probabilities Project http://www.sipta.org/
Summer School on Imprecise Probabilities http://www.idsia.ch/~zaffalon/events/school2004/topics.htm
Society for Risk Analysis Annual Meeting http://www.sra.org/events_2004_meeting.php
Society for Risk Analysis www.sra.org