# First SIPTA School on Imprecise Probabilities

July 27-31, 2004

Lugano, Switzerland # Schedule and Topics

Tuesday 27 July

Speaker: Scott Ferson
.

Title: Introduction to using imprecise probability in risk analysis.

Description: This full-day tutorial workshop introduces five practical and quantitative approaches to risk analysis based on the notions of interval-valued probabilities and imprecisely specified probability distributions. 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. The 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 precise distributions of all the variables in a risk assessment, or the multivariate dependencies among the variables. As a result, in the practical situations where empirical data are limiting, analysts are often forced to make assumptions that can result in assessments that are arbitrarily over-specified and therefore misleading. In practice, the assumptions typically made in these situations, such as independence, (log)normality of distributions, and linear relationships, can under
the risks of adverse events. The assumptions therefore fail to be "protective" in the risk-analytic sense. By relaxing the need to make such unjustified or untenable assumptions, the five approaches based on interval or imprecise probabilities can restore an appropriate degree of conservativism to the analysis.
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. The five approaches based on interval and 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 five approaches also make backcalculations possible and practicable in risk assessments. 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, except by approximate, trial-and-error strategies.
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 presentation style will be casual and interactive. Participants will receive a CD of some demonstration software and the illustrations used during the tutorial.

Overview of topics

What's missing from Monte Carlo?
Correlations are special cases of dependencies
Probability theory has an inadequate model of ignorance
Model uncertainty is epistemic rather than aleatory in nature
Backcalculation cannot be done with Monte Carlo methods

Interval probability
Conjunction and disjunction (ANDs and ORs)
Fréchet case (no assumption about dependence)
Mathematical programming solution
Case study 1: fault-tree for a pressurized tank system
Why and why not use interval probability

Robust Bayes and Bayesian sensitivity analysis
Bayes' rule and the joy of conjugate pairs
Dogma of Ideal Precision
Classes of priors and classes of likelihoods
Robustness and escaping subjectivity
Case study 2: extinction risk and conservation of pinnipeds
Why and why not use robust Bayes

Dempster-Shafer theory
Indistinguishability in evidence
Belief and plausibility
Convolution via the Cartesian product
Case study 3: reliability of dike construction
Case study 4: human health risk from ingesting PCB-contaminated waterfowl
Why and why not use Dempster-Shafer theory

Probability bounds analysis
Marrying interval analysis and probability theory
Fréchet case in convolutions
Case study 5: environmental exposure of wild mink to mercury contaminationbirds to an agricultural insecticide
Backcalculation
Case study 6: planning cleanup for selenium contamination in San Francisco Bay
Why and why not use probability bounds analysis

Imprecise probabilities
Comparative probabilities
Closed convex sets of probability distributions
Multifurcation of the concept of independence
Case study 7: medical diagnosis
Why and why not use imprecise probabilities

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Slides of Scott's talk and exercises are available.

Wednesday 28 July

Speaker:
Gert de Cooman.

Title: Imprecise probability models and their behavioural interpretation.

Description: This tutorial introduces basic notions and ideas in the theory of imprecise probabilities.
There will be a morning and an afternoon session. Both sessions consist of two parts: theory and classroom exercises. In the morning session, we discuss a number of different imprecise probability models: (i) lower and upper previsions; (ii) sets of probability measures; and (iii) sets of desirable gambles. For each of these models we study their interpretation terms of behaviour, the rationality criteria of avoiding sure loss and coherence, and the underlying mechanism, called natural extension, that allows us to make deductions based on these models. We also study their mutual relationships. We show that these models encompass a number of popular uncertainty models and reasoning mechanisms extant in the literature, such as classical propositional logical, Bayesian or precise probabilities, 2-monotone lower probabilities, belief functions, possibility measures.
In the afternoon session, we move on to the notion of conditioning, and shed more light on fundamental results such as the Generalised Bayes Rule and the Marginal Extension Theorem. These lead to techniques, based on the rationality criterion of coherence, that allow us to construct a conditional model from an unconditional one, and to combine conditional and marginal models.
The classroom exercises are intended to allow the students to become more familiar with the more theoretical notions discussed in the theory part.

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Slides of Gert's talk and exercises are available.

Thursday 29 July

Speaker:
Teddy Seidenfeld.

Title: Some decision theory with imprecise and indeterminate probability and utility.
Description:
The day’s sessions will be devoted to understanding how foundational issues in decision theory change when probability (and/or) utility is indeterminate.
We begin with a discussion of canonical expected utility theory, and topics related to decisions in static (non-sequential) decisions. These include criteria relating to coherence, avoiding sure-loss – sometimes called “Book” – and admissibility. We will consider criteria relating to ordering assumptions. Third, we will review results that do not require an Archimedean (or Continuity) condition.
Following that, the discussion will focus on some criteria that affect sequential decision theory, including equivalence of normal and extensive form decisions, and various
notions of “dynamic” coherence.
Next, we will examine what becomes of these same criteria with various decision rules that apply when either probability or utility is allowed to go indeterminate.
The class will include some practice with tools for elicitation and for sequential decision analysis.

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Slides of Teddy's talk are available.

Friday 30 July

Speakers:
Serafín Moral & Marco Zaffalon.

Title: Independence, graphical models, knowledge discovery from data sets under weak assumptions, applications to classification.

Description: First we will review the definition of independence in imprecise probability. As in probability theory, independence is a basic and fundamental concept, but when probabilities are imprecise independence has several possible meanings, and each one of them will have its corresponding formal definition. We will review the different possible interpretations of independence studying their mathematical properties, relationships among them, and pointing at existing open problems. At the same time, we will give urns and coins examples to better understand their practical implications. Finally we will consider the problem of deciding about independence from statistical data. This is a difficult problem, for which classical statistical tests can be applied, but other alternative approaches are also possible. We will show one, based on the measurement of the amount of uncertainty (entropy) of the models, when parameters are inferred with the imprecise Dirichlet model.
Bayesian networks are models to represent complex and uncertain relationships between a large number of variables. They are based on an explicit representation of independence relationships by means of a graph, and procedures to exploit the factorization associated to independence in order to produce fast inferences. They have been very successful in building real applications, but one of their main drawbacks is that, very often it is necessary to give precise estimations for a large number of probability values, sometimes with very small sample sizes. Credal networks try to avoid this difficulty by allowing the use of imprecise probabilities. We will review the work that has been done in credal networks with two main points: inference (much more difficult than with precise probability) and learning from data (in general, probabilistic procedures are applied to learn the structure and very few genuine methods based on imprecision have been proposed).
More generally speaking, we will show that the task called knowledge discovery from data sets can benefit from adopting imprecise probability methods. Knowledge discovery typically assumes that data are the only source of information about a domain, and aims at inferring models that make domain knowledge explicit. Learning from data is thus started in conditions of prior ignorance; and the data are often available in incomplete way, such as when values are missing in the data set, which involves another form of ignorance that is about the data themselves. When pattern classification is concerned, the inferred models are used in practice to do medical diagnosis, fraud detection, or image recognition, just to name a few applications. Modeling ignorance carefully is a central issue to make these models and applications reliable. This issue is strictly related to the possibility to state, and work with, weak assumptions.
Initially we will show how imprecise probability allows to reliably dealing with incomplete data in a way that significantly departs from established approaches. Missing data are a serious problem of knowledge discovery application, that can severely limit the credibility of the inferred models. Imprecise probability makes robust modeling of missing data possible by permitting to do no assumptions on the mechanism that turns complete into incomplete data. The issues of learning from, and classifying, incomplete data will be treated in a unified framework by a generalized updating rule. This will naturally produce generalized classifiers, called credal classifiers, with the novel characteristic of being able to (partially) suspend the judgment when there are reasonable doubts about the correct classification. Credal classifiers will be shown to be able to carefully deal also with the prior ignorance problem, by relying on the imprecise Dirichlet model.
Finally, we will focus on the practical design of credal classifiers. We will consider the naive Bayes, TAN, and C4.5 classifiers. Naive Bayes and TAN are special cases of Bayesian networks, while C4.5 is a classification tree. These are traditional classifiers, which are very popular and widely recognized to be good in the knowledge discovery community. We will review the extension of these models to credal classification, showing how to infer them from data and to carry out the classification. Real case studies will be presented to show the impact of credal classification.

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Slides of Serafin's talk and exercises are available.

Slides of Marco's talk and exercises are available.

Saturday 31 July

Speaker:
Thomas Augustin.

Title: Robust Neyman Pearson theory & summary view on imprecise probabilities.

Description: The last lecture will consist of two parts. The first part is dedictated to hypotheses testing with imprecise probabilities. We first motivate, explore and generalize several neighborhood models popular in robust statistics. Then the Huber-Strassen theory is reviewed and extended. Finally, we discuss algorithms to construct optimal testing (and decision) procedures and give a brief application to optimal investment strategies based on robust utility functionals.
The second part tries to review and summarize the previous lectures of the school, with the aim to repeat, unify and fix the most important techniques and results.

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Slides of Thomas' talk and summary lecture are available.

Typical schedule

08:30-10:30 Talk
10:30-11:00 Coffee break

11:00-13:00 Exercises
13:00-14:30 Lunch

14:30-16:30 Talk
16:30-17:00 Coffee break
17:00-19:00 Exercises