Imprecise Probabilities

This chapter explores the topic of imprecise probabilities (IP) as it re-lates to model validation. IP is a family of formal methods that aim to provide a better representation of severe uncertainty than is possible with standard probabilistic methods. Among the methods discussed here are using sets of probabilities to represent uncertainty, and using functions that do not satisfy the additvity property. We discuss the basics of IP, some examples of IP in computer simulation contexts, possible interpretations of the IP framework, and some conceptual problems for the approach. We conclude with a discussion of IP in the context of model validation. be valuable reason to to to of (or a relation

[1]  David Ríos Insua,et al.  Robust Bayesian analysis , 2000 .

[2]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[3]  Peter Walley.  A Bounded Derivative Model for Prior Ignorance about a Real-valued Parameter , 1997 .

[4]  Peter Walley,et al.  Analysis of Clinical Data Using Imprecise Prior Probabilities , 1996 .

[5]  S. Ferson,et al.  Different methods are needed to propagate ignorance and variability , 1996 .

[6]  Peter Walley,et al.  Measures of Uncertainty in Expert Systems , 1996, Artif. Intell..

[7]  L. Wasserman,et al.  Inferences from multinomial data: Learning about a bag of marbles - Discussion , 1996 .

[8]  P. Walley.  Inferences from Multinomial Data: Learning About a Bag of Marbles , 1996 .

[9]  James O. Berger,et al.  An overview of robust Bayesian analysis , 1994 .

[10]  Larry Wasserman,et al.  Dilation for Sets of Probabilities , 1993 .

[11]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[12]  Paul Krause,et al.  Representing Uncertain Knowledge , 1993, Springer Netherlands.

[13]  P. Walley,et al.  Robust Bayesian credible intervals and prior ignorance , 1991 .

[14]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[15]  Glenn Shafer,et al.  Perspectives on the theory and practice of belief functions , 1990, Int. J. Approx. Reason..

[16]  I. Graham.  Non-standard logics for automated reasoning , 1990 .

[17]  Luis M. de Campos,et al.  The concept of conditional fuzzy measure , 1990, Int. J. Intell. Syst..

[18]  P. Walley,et al.  Uncertainty and indeterminacy in assessing the economic viability of energy options; A case study of solar heating systems in Brazil , 1990 .

[19]  Max Henrion,et al.  Uncertainty: A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis , 1990 .

[20]  Joseph B. Kadane,et al.  You have printed the following article : On the Shared Preferences of Two Bayesian Decision Makers , 2008 .

[21]  M. Smithson.  Ignorance and Uncertainty , 1989, Cognitive Science.

[22]  Teddy Seidenfeld,et al.  Decision Theory Without “Independence” or Without “Ordering” , 1988, Economics and Philosophy.

[23]  Terrence L. Fine,et al.  Lower probability models for uncertainty and nondeterministic processes , 1988 .

[24]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[25]  Ronald P. Loui,et al.  Decisions with indeterminate probabilities , 1986 .

[26]  I. Levi.  Hard Choices: Decision Making under Unresolved Conflict , 1986 .

[27]  R. Hogarth,et al.  Ambiguity and Uncertainty in Probabilistic Inference. , 1985 .

[28]  Lotfi A. Zadeh,et al.  Is Probability Theory Sufficient for Dealing with Uncertainty in AI: A Negative View , 1985, UAI.

[29]  J. Hartigan.  Theories of Probability , 1983 .

[30]  T. Fine,et al.  Towards a Frequentist Theory of Upper and Lower Probability , 1982 .

[31]  J. Hartigan,et al.  Bayesian Inference Using Intervals of Measures , 1981 .

[32]  Martin Edman.  The Probable and the Provable , 1980 .

[33]  Isaac Levi,et al.  The Enterprise Of Knowledge , 1980 .

[34]  Francisco Javier Girón González-Torre,et al.  Quasi-Bayesian behaviour: a more realistic approach to decision making? , 1980 .

[35]  Terrence L. Fine.  An Argument for Comparative Probability , 1977 .

[36]  R. Giles.  A Logic for Subjective Belief , 1976 .

[37]  I. Levi.  On Indeterminate Probabilities , 1974 .

[38]  B. Sauphanor.  The logical foundations of statistical inference , 1974 .

[39]  Henry E. Kyburg,et al.  Probability and the logic of rational belief , 1970 .

[40]  A. Dempster.  Upper and Lower Probabilities Generated by a Random Closed Interval , 1968 .

[41]  A. Alexandrova.  The British Journal for the Philosophy of Science , 1965, Nature.

[42]  Irving John Good,et al.  Subjective Probability as the Measure of a Non-measurable Set , 1962 .

[43]  D. Ellsberg.  Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[44]  Cedric A. B. Smith,et al.  Consistency in Statistical Inference and Decision , 1961 .

[45]  G. Choquet.  Theory of capacities , 1954 .

[46]  B. O. Koopman.  The bases of probability , 1940 .

[47]  H. Jeffreys.  A Treatise on Probability , 1922, Nature.