What Is Imprecision?

Uncertainty is usually modelled by a probability distribution, and treated using techniques from probability theory. Such an uncertainty model will often be inadequate in cases where insufficient information is available to identify a unique probability distribution. In that case, imprecise probabilities aim to represent and manipulate the really available knowledge about the system.

Similar concerns arise when dealing with utility. In making decisions, each reward is assigned a single real number, called utility, and rewards are accordingly ranked. However, in many practical cases, a complete ranking over all rewards is unrealistic. Imprecise utility aims to represent and reason with such incomplete preferences over rewards.

The main benefits of using imprecise probabilities and imprecise utilities, compared to classical statistical methods, are

• more reliable inference
• no pressing need for sensitivity analysis, as this is built into the model itself
• indecision can be explicitly modelled
• effect of modelling assumptions and information on inference is more apparent and credible
• information from different sources can be coherently combined

The term imprecision actually covers a very wide range of extensions of the classical theory of probability. To sum just a few, they include

• lower and upper previsions (Walley, 1991)
• belief functions (Dempster, 1967; Shafer, 1976), theory of hints (Kohlas and Monney, 1995), transferable belief model (Smets, 1992)
• possibility measures (Dubois, 1985, 1988)
• credal sets and sets of probabilities and utilities (Levi, 1980)
• risk measures (Artzner et al., 1999)
• 2- and n-monotone set functions, Choquet capacities (Choquet, 1953)
• comparative probability orderings (Keynes, 1921; De Finetti, 1931; Fine, 1973; also see overview by Fishburn, 1986)
• robust Bayes methods (Berger, 1984)
• sets of desirable gambles (Walley, 1991)
• p-boxes (Ferson et al., 2003)
• lower and upper envelopes/collectives (Papamarcou and Fine, 1991)
• interval probability (Weichselberger 2000, 2001)
• capacities (Huber, 1965; Huber and Strassen, 1973)
• ambiguity (Ellsberg, 1961)
• logical/fiducial probabilities (Hampel, 1993; Weichselberger, 2005)
• linear partial information (Kofler and Menges, 1976)
• multiple priors (Gilboa, 1989)
• partial identification of probability distributions (Manski, 2003)

Recent Advances in Statistics Using Imprecision

Currently, a popular approach to statistics using imprecision, is by use of Walley's generalised Bayes rule, which is close in nature to the robust Bayesian approach where a set of priors is used, and each prior in the set is updated to produce a set of posteriors. A particularly successful model, be it not without its critics, is the imprecise Dirichlet model, which has been applied for example in game theory, classification, Markov decision processes, aggregation, etc. Other statistical methods with promising potential for application include the bounded derivative model, and nonparametric predictive inference.

In classical statistics, limit theorems play a crucial role. Recently, generalised versions of such theorems have been presented, opening new possibilities from a frequentist perspective.

Imprecise probabilities and utilities also lead the way to generalised decision support methods. For example, generalisations of maximising expected utility lead to interesting research challenges, the solutions to some of which have been presented and addressed. A common theme in all of these generalised methods, is that they focus on realistic reflection of available information and preferences, and as such support informative decisions. This also poses challenges for elicitation and data collection.

Aim Of The Special Issue

With this special issue on imprecision we hope to promote new and recent techniques that employ imprecise methods in a useful way, and advance them to a wider audience. We especially hope to demonstrate the benefits of imprecise models over traditional statistical methods. In particular we are looking for (but not exclusively):

• Applications enhanced by use of imprecision (less information, fewer assumptions).
• Theoretical and methodological developments inspired by practical problems, and illustrating their use in such problems.
• Studies, with examples of practical nature, to emphasise advantages and disadvantages of imprecise methods compared to classical (both frequentist and Bayes) inferential methods, and also to show similarities and differences to robust statistical and nonparametric methods (see above list).
• Decision support using imprecision in probabilities and/or utilities, with applications or illustrations from a practical perspective.

References

• J. Abellán and S. Moral. Upper entropy of credal sets. Applications to credal classification. International Journal of Approximate Reasoning, 39:235-255, 2005.
• Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath. Coherent measures of risk. Mathematical Finance, 9(3):203-228, 1999.
• Thomas Augustin and Frank P. A. Coolen. Nonparametric predictive inference and interval probability. Journal of Statistical Planning and Inference, 124:251-272, 2004.
• James O. Berger. The robust Bayesian viewpoint. In J. B. Kadane, editor, Robustness of Bayesian Analyses, pages 63-144. Elsevier Science, Amsterdam, 1984.
• Jean-Marc Bernard. An introduction to the imprecise Dirichlet model for multinomial data. International Journal of Approximate Reasoning, 39(2-3):123-150, 2005.
• G. Choquet. Theory of capacities. Annales de l'Institut Fourier, 5:131-295, 1953-54.
• Frank P. A. Coolen and Pauline Coolen-Schrijner. Nonparametric predictive comparison of proportions. Journal of Statistical Planning and Inference, 137:23-33, 2007.
• Pauline Coolen-Schrijner and Frank P. A. Coolen. Adaptive age replacement strategies based on nonparametric predictive inference. Journal of the Operational Research Society, 55:1281-1297, 2004.
• Gert de Cooman and Marco Zaffalon. Updating beliefs with incomplete observations. Artificial Intelligence, 159(1-2):75-125, 2004.
• Gert de Cooman and Matthias C. M. Troffaes. Dynamic programming for deterministic discrete-time systems with uncertain gain. International Journal of Approximate Reasoning, 39(2-3):257-278, Jun 2005.
• Gert de Cooman and Matthias C. M. Troffaes. Dynamic programming for deterministic discrete-time systems with uncertain gain. International Journal of Approximate Reasoning, 39(2-3):257-278, Jun 2005.
• A. P. Dempster. Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist., 38:325-339, 1967.
• Dieter Denneberg. Non-additive Measure and Integral. Kluwer, Dordrecht, 1994.
• Didier Dubois and Henri Prade. Possibility Theory - An Approach to Computerized Processing of Uncertainty. Plenum Press, New York, 1988.
• Daniel Ellsberg. Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 75(4):643-669, 1961.
• Scott Ferson, Vladik Kreinovich, Lev Ginzburg, Davis S. Myers, and Kari Sentz. Constructing probability boxes and Dempster-Shafer structures. Technical Report SAND2002-4015, Sandia National Laboratories, January 2003.
• Terrence L. Fine. Lower probability models for uncertainty and nondeterministic processes. Journal of Statistical Planning and Inference, 20:389-411, 1988.
• P. C. Fishburn. The axioms of subjective probability. Statistical Science, 1:335-358, 1986.
• Itzhak Gilboa and David Schmeidler. Maxmin expected utility with non-unique prior. Journal of Mathematical Economics, 18(2):141-153, 1989.
• F. Hampel. Some thoughts about the foundations of statistics. In S. Morgenthaler, E. Ronchetti, and W. A. Stahel, editors, New Directions in Statistical Data Analysis and Robustness, pages 125-137. Birkhäuser, Basel, 1993.
• Peter J. Huber. A robust version of the probability ratio test. The Annals of Mathematical Statistics, 36(6):1753-1758, 1965.
• Peter J. Huber. The use of Choquet capacities in statistics. Bulletin of the International Statistical Institute, XLV, Book 4, 1973.
• Peter J. Huber and Volker Strassen. Minimax tests and the Neyman-Pearson lemma for capacities. The Annals of Statistics, 1(2):251-263, 1973.
• E. Kofler and G. Menges. Entscheidungen bei unvollständiger Information, volume 136 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, 1976.
• E. Kofler, Z. W. Kmietowicz, and A. D. Pearman. Decision making with linear partial information (L.P.I.). The Journal of the Operational Research Society, 35(12):1079-1090, 1984.
• J. Kohlas and P.-A. Monney. Mathematical Theory of Hints (An Approach to the Dempster-Shafer Theory of Evidence), volume 425 of Lecture Notes in Economics and Mathematical Systems. Springer, 1995.
• Isaac Levi. On indeterminate probabilities. Journal of Philosophy, 71:391-418, 1974.
• Isaac Levi. The Enterprise of Knowledge. An Essay on Knowledge, Credal Probability, and Chance. MIT Press, Cambridge, 1980.
• Charles Manski. Partial Identification of Probability Distributions. Springer Series in Statistics. Springer, New York, 2003.
• Adrian Papamarcou and Terrence L. Fine. Unstable collectives and envelopes of probability measures. The Annals of Probability, 19(2):893-906, 1991.
• R. Pelessoni and P. Vicig. Coherent risk measures and upper previsions. In G. de Cooman, T. L. Fine, and T. Seidenfeld, editors, ISIPTA '01 - Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, pages 307-315, Maastricht, 2001. Shaker Publishing.
• Glenn Shafer. A Mathematical Theory of Evidence. Princeton University Press, 1976.
• P. Smets. Resolving misunderstandings about belief functions. International Journal of Approximate Reasoning, 6:321-344, 1992.
• Matthias C. M. Troffaes. Learning and optimal control of imprecise Markov decision processes by dynamic programming using the imprecise Dirichlet model. In Miguel Lopéz-Díaz, María Á. Gil, Przemyslaw Grzegorzewski, Olgierd Hyrniewicz, and Jonathan Lawry, editors, Soft Methodology and Random Information Systems, pages 141-148, Berlin, 2004. Springer.
• Matthias C. M. Troffaes. Decision making under uncertainty using imprecise probabilities. International Journal of Approximate Reasoning, 45:17-29, 2007.
• Lev V. Utkin and Thomas Augustin. Decision making under incomplete data using the imprecise Dirichlet model. International Journal of Approximate Reasoning, 44:322-338, 2007.
• Peter Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.
• Peter Walley. Inferences from multinomial data: Learning about a bag of marbles. Journal of the Royal Statistical Society, 58(1):3-34, 1996.
• Peter Walley. A bounded derivative model for prior ignorance about a real-valued parameter. Scandinavian Journal of Statistics, 24(4):463-483, 1997.
• K. Weichselberger. The theory of interval probability as a unifying concept for uncertainty. International Journal of Approximate Reasoning, 24:149-170, 2000.
• K. Weichselberger. Elementare Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung I - Intervallwahrscheinlichkeit als umfassendes Konzept. Physica, Heidelberg, 2001. In cooperation with T. Augustin and A. Wallner.
• K. Weichselberger. The logical concept of probability and statistical inference. In Fabio G. Cozman, Robert Nau, , and Teddy Seidenfeld, editors, ISIPTA '05: Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, pages 396-405, Pittsburgh, USA, July 2005.
• Marco Zaffalon. The naive credal classifier. Journal of Statistical Planning and Inference, 105(1):5-21, June 2002.
• Marco Zaffalon, Keith Wesnes, and Orlando Petrini. Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data. Artificial Intelligence in Medicine, 29(1-2):61-79, 2003.