Programme

Day Subject Led by
Mon 12 May research presentations
Tue 13 May regression methods Jochen Einbeck, Frank Coolen, and Gero Walter
Wed 14 May Markov theory Damjan Skulj and Marco Cattaneo
Thu 15 May decision making Matthias Troffaes and Robert Hable
Fri 16 May principles and methods of statistics Frank Coolen and Thomas Augustin

Monday Presentations

Time Event/Speaker Abstract
10:00 Coffee
10:20 Glenn Shafer The Wiki of the online prediction group (Royal Holloway), and some open problems in probability theory.
11:10 Sebastien Destercke Relating independence notions of imprecise probability theory to event trees. There are currently two main approaches to probability theory that uses lower and upper expectations: Walley's behavioural approach, and Shafer and Vovk's game-theoretic framework, where event trees play a central role. De Cooman and Hermans have shown that these two approaches can be related to each other, and they have introduced imprecise probability trees as a bridge between them, making a step towards a more unified handling of uncertainty. In this talk, we go a little step further into such a unification, by giving first results relating notion of independence originating from Walley's behavioral theory to the notion of event trees independence given by Shafer. In the first part of the talk, we give an account of recent results showing that the notion of forward irrelevance is equivalent to event-tree independence in particular event trees. We argue that, in a theory of uncertain processes, the asymmetrical notion of epistemic irrelevance has a more important role to play than its more involved and symmetrical counterpart called epistemic independence. In the second part of the talk, we give some preliminary ideas as to how the more involved notion of epistemic independence could be related to event trees, and raise the issues of relating other structural judgments such as permutability or exchangeability in event trees.

Results presented and discussed in this talk are the consequences of a joint work and of fruitful discussions with Gert de Cooman, most of them done during a two month stay at SYSTeM research group.

References

  • De Cooman G, Hermans F (2007) On coherent immediate prediction: Connecting two theories of imprecise probability. In: De Cooman G, Vejnarova J, Zaffalon M (eds) ISIPTA '07 - Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications, SIPTA, pp. 107-116
  • De Cooman G, Hermans F (2008) Imprecise probability trees: Bridging two theories of imprecise probability. Artificial Intelligence DOI 10.1016/j.artint.2008.03.001, in press
  • De Cooman G, Miranda E (2008) Forward irrelevance. Journal of Statistical Planning and Inference, DOI 10.1016/j.jspi.2008.01.012, in press
  • Shafer G (1996) The Art of Causal Conjecture. The MIT Press, Cambridge, MA
  • Shafer G, Vovk V (2001) Probability and Finance: It’s Only a Game! Wiley, New York
  • Walley P (1991) Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London
12:00 Lunch
14:00 Ian Vernon Galaxies and imprecision.
14:50 Carolin Strobl Classification - An Overview over Standard and Credal Classification Approaches with an Emphasis on Instability and Interpretability. Classification trees are one representative out of a vast variety of statistical classification techniques. While their nonparametric and flexible recursive partitioning approach has added much to the popularity of classification trees in many applied sciences, they are known to be instable with respect to small changes in the learning data and their variable selection criteria can be artificially biased in favor of variables of certain types.

Different advancements of the standard classification tree methodology have been suggested to deal with the issue of instability, as well as the related issue of overfitting: The ensemble methods bagging and random forests have been suggested in the machine learning community and employ random sampling techniques to generate sets of classification trees. Another, more recent and thus less well known approach from this community is TWIX, where sets of classification trees are generated by splitting in additional optimal cutpoints. In the imprecise probabilities community, on the other hand, credal classification trees have been proposed, that employ an upper entropy approach for conservative split selection in order to avoid overfitting.

The talk gives a short overview over these existing methods with an emphasis on the issues of interpretability and unbiased variable selection. For the TWIX approach an adaptive, data-driven split selection criterion is suggested, that turns out to be closely related to the Imprecise Dirichlet Model. With this criterion, additional cutpoints are selected in a data-driven way to create a set of trees, while the ensemble reduces to a single, interpretable tree when the partition is sufficiently stable.
15:40 Tea
16:10 Marco Cattaneo A hierarchical model based on the likelihood function. If we interpret the statistical likelihood function as a measure of the relative plausibility of the probabilistic models considered, then we obtain a hierarchical description of uncertain knowledge, offering a unified approach to the combination of probabilistic and possibilistic uncertainty. The fundamental advantage of the resulting fuzzy probabilities with respect to imprecise probabilities is the ability of using all the information provided by the data.

Tuesday: Regression Methods

Time Speaker Title
9:15 Jochen Einbeck Regression, robustness, imprecision, and 'trust'.
10:00 Coffee break
10:15 Gero Walter Linear regression analysis under sets of conjugate priors
11:00 Mike Lonergan Are harbour seal populations declining? The problem of detecting changes in complex systems from small numbers of noisy samples.
11:20 Discussions
12:00 Lunch
14:00 Thomas Augustin Imprecision and random effects
Robert Hable Robust regression
Carolin Strobl Regression and classification

Wednesday: Markov Theory

Time Speaker Title
10:00 Damjan Skulj Computational problems in the theory of Markov chains with interval probabilities.

A Markov chain model is presented where precise probabilities are replaced with interval probabilities. The generalisation allows modelling situations where transition probabilities are allowed to vary within given intervals. Consequently, the distributions corresponding to further stages and limiting distributions are also sought in terms of interval probabilities. The calculations of the intervals corresponding to the steps of generalised Markov chains are computationally very complex, and therefore more efficient methods should be sought. A class of so called regular Markov chains with interval probabilities behaves in a similar ways as classical regular Markov chains. Another open problem is how to efficiently characterise regularity in terms of interval probabilities.
11:00 Break
11:15 Gert De Cooman Imprecise Markov chains and their limit behaviour.

When the parameters of a finite Markov chain in discrete time, i.e., its initial and transition probabilities, are not well known, we can and should perform a sensitivity analysis. This is done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. I will show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. I will also indicate how the inferred credal set about the state at time n evolves as n goes to infinity, and show that under quite unrestrictive conditions, this credal set converges to a uniquely invariant credal set, regardless of the credal set given for the initial state of the system. This effectively leads to a Perron-Frobenius Theorem for a special class of non-linear dynamical systems in discrete time.
12:30 Lunch
14:00 Richard Crossman Long-Term Behaviour of Imprecise Markov Chains

In recent years work has begun upon considering the long-term behaviour of Markov Chains for which the elements of the transition matrix at a given time step n are not precisely known, nor assumed to be independent of time, but are known to exist within given bounds. However, in situations where the chain is known to contain an absorbing state (that is a state for which the lower probability of leaving the state is equal to 1), we can show that the long-term behaviour is often just what we would expect, i.e. certain absorption. In this talk, then, we consider how to condition upon non-absorption for the case of imprecise Markov chains.
15:00 Break
15:15 Filip Hermans

Thursday: Decision Making

Time Speaker Title
9:30 Robert Hable An introduction to decision making with imprecise probabilities.
10:10 Thomas Augustin Computational Aspects of Decision Making
10:30 Coffee
11:00 Marco Cattaneo Statistical Decisions Based Directly on the Likelihood Function
11:30 Malcolm Farrow and Michael Goldstein Almost-Pareto decision sets in imprecise utility hierarchies.

We develop methods for analysing decision problems based on multi-attribute utility hierarchies, structured by mutual utility independence, in which trade-offs between the various attributes are not precisely specified. Instead, our analysis is based on whatever limited collection of preferences we may assert between attribute collections. These preferences identify a class of Pareto optimal decisions. We show how to reduce the class further by combining rules which are almost equivalent and introduce general principles appropriate to selecting decisions in an imprecise hierarchy. The approach is illustrated by the design of a university course module.
12:30 Lunch
14:00 Matthias Troffaes and Nathan Huntley Some issues in sequential decision making.

In this talk we will present a method for solving decision trees under Walley's maximality criterion. We discuss how maximality potentially leads to counterfactual decision sets, because a decision that is optimal in a subtree can be potentially absent in any optimal policy in the full tree.
14:30 Thomas Augustin A note on updating in decision theory.
15:00 Break
15:30 Peter Matthews A Prototype Implementation of Bayesian Belief Networks for Engineering Design Decision Support.

Engineering Design is a complex process wherein the fluidity of the domain during the earliest phases of the process resists deterministic models. To overcome this, a Bayesian Belief Network was induced for a given design domain, thereby providing a more fluid and dynamic stochastic domain representation. A prototype user interface for this BBN was developed using Visual Basic in Microsoft Excell, and this was subsequently tested with two different user groups and two design domains: (1) a set of student designers considering a conceptual car domain and (2) a team of industrial designers considering a gas turbine combustor. The key challenge identified during these trials was the understanding of the domain through a BBN and presentation of the BBN user interface.

Friday: Principles and Methods of Statistics

Time Speaker Title
9:30 Frank Coolen and Marco Cattaneo Updating from set-valued observations
10:30 Coffee
10:45 Gero Walter Handling prior-data conflict in generalized Bayesian updating
11:20 Frank Coolen and Thomas Augustin Generalized Bayes rule and beyond
12:30 Lunch
14:00 Matthias Troffaes and Gero Walter Software and simulation with imprecise probabilities
15:00 Everyone Looking back and future plans
16:00 Everyone Having a drink in town

Confirmed Participants