Skip to main content
SpringerLink
Log in

Quasi-Bayesian Analysis Using Imprecise Probability Assessments And The Generalized Bayes’ Rule

  • Published:
Theory and Decision Aims and scope Submit manuscript

Abstract

The generalized Bayes’ rule (GBR) can be used to conduct ‘quasi-Bayesian’ analyses when prior beliefs are represented by imprecise probability models. We describe a procedure for deriving coherent imprecise probability models when the event space consists of a finite set of mutually exclusive and exhaustive events. The procedure is based on Walley’s theory of upper and lower prevision and employs simple linear programming models. We then describe how these models can be updated using Cozman’s linear programming formulation of the GBR. Examples are provided to demonstrate how the GBR can be applied in practice. These examples also illustrate the effects of prior imprecision and prior-data conflict on the precision of the posterior probability distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • K.A. Andersen J.N. Hooker (1994) ArticleTitleBayesian logic Decision Support Systems 11 191–210 Occurrence Handle10.1016/0167-9236(94)90031-0

    Article  Google Scholar 

  • R.W. Becker F.O. Brownson (1964) ArticleTitleWhat prmbig uity? or the role of ambiguity in decision making Journal of Political Economy 72 62–73 Occurrence Handle10.1086/258854

    Article  Google Scholar 

  • P.G. Benson K.M. Whitcomb (1993) ArticleTitleThe effectiveness of imprecise probability forecasts Journal of Forecasting 12 139–159

    Google Scholar 

  • J.O. Berger (1985) Statistical Decision Theory and Bayesian Analysis EditionNumber2 Springer-Verlag New York

    Google Scholar 

  • R. Beyth-Marom (1982) ArticleTitleHow probable is probable? a numerical translation of verbal probability expressions Journal of Forecasting 1 257–269

    Google Scholar 

  • L.G. Boiney (1993) ArticleTitleThe effects of skewed probability on decision making under ambiguity Organizational Behavior and Human Decision Processes 56 134–148 Occurrence Handle10.1006/obhd.1993.1048

    Article  Google Scholar 

  • Breese, J.S. and Fertig, K.W. (1991), Decision making with interval influence diagrams. In Bonissone, P.P., Henrion, M., Kanal, L.N., and Lemmer, J.F. (eds.), Uncertainty in Artificial Intelligence 6, Elsevier Science Publishers, pp. 467–478.

  • G. Coletti (1994) ArticleTitleCoherent Numerical and Ordinal Probabilistic Assessments IEEE Transactions on Systems, Man, and Cybernetics 24 1747–1753

    Google Scholar 

  • Coletti, G. and Scozzafava, R. (1999), Coherent upper and lower bayesian updating, in G. DeCooman, F.G. Cozman, S. Moral and Walley, P. (eds.), Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, pp. 101–110, The Imprecise Probabilities Project.

  • F.P.A. Coolen (1992) Elicitation of expert knowledge and assessment of imprecise prior densities for lifetime distributions, Memorandum COSOR 92–12 Eindhoven University of Technology, Department of Mathematics and Computing Science Eindhoven, The Netherlands

    Google Scholar 

  • F.P.A. Coolen (1993) ArticleTitleImprecise conjugate prior densities for the one-parameter exponential family of distributions Statistics and Probability Letters 16 337–342 Occurrence Handle10.1016/0167-7152(93)90066-R Occurrence HandleMR1225075

    Article  MathSciNet  Google Scholar 

  • F.P.A. Coolen M.J. Newby (1994) ArticleTitleBayesian reliability analysis with imprecise prior probabilities Reliability Engineering and System Safety 43 75–85 Occurrence Handle10.1016/0951-8320(94)90096-5

    Article  Google Scholar 

  • F.G. Cozman (1997) An informal introduction to quasi-bayesian theory for AI Carnegie Mellon University Pittsburgh, PA

    Google Scholar 

  • Cozman, F.G. (1999), Computing posterior upper expectations, in G. DeCooman, F.G. Cozman, S. Moral and Walley, P. (eds.), Proceedings of the First International Symposium on Imprecise Probabilities and Their Applications, The Imprecise Probabilities Project, pp. 131–140.

  • F.G. Cozman (2002) Algorithms for conditioning on events of zero lower probability. Proceedings of the Fifteenth International Florida Artificial Intelligence Society Pensacola Florida 248–252

    Google Scholar 

  • F.H.J. Crome M.R. Thomas L.A. Moore (1996) ArticleTitleA novel approach to assessing impacts of rain forest logging Ecological Applications 6 104–123

    Google Scholar 

  • S.P. Curley J.F. Yates (1985) ArticleTitleThe center and range of probability intervals as factors affecting ambiguity preferences Organizational Behavior and Human Decision Processes 36 273–287 Occurrence Handle10.1016/0749-5978(85)90016-0

    Article  Google Scholar 

  • Dickey, J. (2003), Convenient interactive computing for coherent imprecise prevision assessments, in J.M. Bernard, T. Seidenfeld and M.␣Zaffalon (eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications,pp. 218–230, Carleton Scientific.

  • H.J. Einhorn R.M. Hogarth (1985) ArticleTitleAmbiguity and uncertainty in probabilistic inference Psychological Review 92 433–461 Occurrence Handle10.1037/0033-295X.92.4.433

    Article  Google Scholar 

  • D. Ellsberg (1961) ArticleTitleRisk, uncertainty, and the savage axioms Quarterly Journal of Economics 75 643–669

    Google Scholar 

  • E. Fagiuoli M. Zaffalon (1998) ArticleTitle2U: An exact interval propagatgion algorithm for polytrees with binary variables Artificial Intelligence 106 77–107 Occurrence Handle10.1016/S0004-3702(98)00089-7

    Article  Google Scholar 

  • P.M. Fayers D. Ashby M.K.B. Parmar (1997) ArticleTitleTutorial in biostatistics: bayesian data monitoring in clinical trials Statistics in Medicine 16 1413–1430 Occurrence Handle10.1002/(SICI)1097-0258(19970630)16:12<1413::AID-SIM578>3.3.CO;2-L Occurrence Handle9232762

    Article  PubMed  Google Scholar 

  • T.B. Feagans W.F. Biller (1981) ArticleTitleAssessing the health risks associated with air quality standards The Environmental Professional 3 235–247

    Google Scholar 

  • K.W. Fertig J.S. Breese (1990) Interval influence diagrams M. Henrion R.D. Shachter L.N. Kanal J.F. Lemmer (Eds) Uncertainty in Artificial Intelligence 5 Elsevier Science Publishers Amsterdam, North-Holland 149–161

    Google Scholar 

  • T.L. Fine (1988) ArticleTitleLower probability models for uncertainty and nondeterministic processes Journal of Statistical Planning and Inference 20 389–411 Occurrence Handle10.1016/0378-3758(88)90099-7 Occurrence HandleMR976185

    Article  MathSciNet  Google Scholar 

  • Fortin, V., Parent, E. and Bobée, B. (2001), Posterior previsions for the parameter of a binomial model via natural extension of a finite number of judgments, in G. DeCooman, T. Fine and Seidenfeld, T. (eds.),Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing.

  • P. Gardenfors N.E. Sahlin (1982) ArticleTitleUnreliable probabilities, risk taking, and uncertainty Synthese 53 361–386 Occurrence Handle10.1007/BF00486156

    Article  Google Scholar 

  • R.W. Goldsmith N.E. Sahlin (1982) The Role of second-order probabilities in decision making P.C. Humphreys O. Svenson A. Vari (Eds) Analysing and Aiding Decision Processes North Holland Amsterdam 319–329

    Google Scholar 

  • I.J. Good (1965) The Estimation of Probabilities: An Essay on Modern Bayesian Methods MIT Press Cambridge, Massachusetts

    Google Scholar 

  • S.H. Hurlbert (1984) ArticleTitlePseudoreplication and the design of ecological field experiments Ecological Monographs 54 187–211

    Google Scholar 

  • B.O. Koopman (1940) ArticleTitleThe axioms and algebra of intuitive probability Annals of Mathematics 41 269–292

    Google Scholar 

  • Kozine, I. and Krymsky, V. (2003), Reducing uncertainty by imprecise judgments on probability distribution: application to system reliability. in J.M. Bernard, T. Seidenfeld and Zaffalon, M. (eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, pp. 335–344, Carleton Scientific.

  • I. Levi (1980) The Enterprise of Knowledge Cambridge MIT Press Massachusetts

    Google Scholar 

  • Lins, G.C.N. and Campello de Souza, F.M. (2001), A protocol for the elicitation of prior distributions, in G. DeCooman, T. Fine and Seidenfeld, T. (eds.), Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing.

  • H.F. Martz R.A. Waller (1982) Bayesian Reliability Analysis John Wiley and Sons New York

    Google Scholar 

  • S. Moral J. Sagnado Particledel (1997) Aggregation of imprecise probabilities B. Bouchon-Meunier (Eds) Aggregation and Fusion of Imperfect Information. Physica-Verlag Heidelberg Germany 162–188

    Google Scholar 

  • M.K.B. Parmar D.J. Spiegelhalter L.S. Freedman (1994) ArticleTitleThe CHART trials: bayesian design and monitoring in practice Statistics in Medicine 13 1297–1312 Occurrence Handle7973211

    PubMed  Google Scholar 

  • L.J. Savage (1954) The Foundations of Statistics John Wiley and Sons New York

    Google Scholar 

  • Schervish, M.J., Seidenfeld, T., Kadane, J.B. and Levi, I. (2003), Extensions of expected utility and some limitations of pairwise comparisons, in J.M. Bernard, T. Seidenfeld and Zaffalon, M. (eds.), Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications, pp. 496–510, Carleton Scientific.

  • T. Seidenfeld L. Wasserman (1993) ArticleTitleDilation for sets of probabilities Annals of Statistics 21 1139–1154

    Google Scholar 

  • C.A.B. Smith (1961) ArticleTitleConsistency in statistical inference and decision Journal of the Royal Statistical Society B 23 1–25

    Google Scholar 

  • P. Snow (1991) ArticleTitleImproved posterior probability estimates from prior and conditional linear constraint systems IEEE Transactions on Systems, Man and Cybernetics Part A 21 464–469

    Google Scholar 

  • D.J. Spiegelhalter L.S. Freedman M.K.B. Parmar (1993) ArticleTitleApplying bayesian thinking in drug development and clinical trials Statistics in Medicine 12 1501–11 Occurrence Handle8248659

    PubMed  Google Scholar 

  • B. Tessem (1992) ArticleTitleInterval probability propagation International Journal of Approximate Reasoning 7 95–120 Occurrence Handle10.1016/0888-613X(92)90006-L

    Article  Google Scholar 

  • Utkin, L.V. and Kozine I.O. (2001), Computing the reliability of complex systems. in G. DeCooman, T. Fine and Seidenfeld, T. (eds.), Proceedings of the Second Internation Symposium on Imprecise Probabilities and Their Applications, Shaker Publishing.

  • L.V. Utkin S.V. Gurov (2002) ArticleTitleImprecise reliability measures for some new lifetime distribution classes Journal of Statistical Planning and Inference 105 215–232 Occurrence Handle10.1016/S0378-3758(01)00211-7

    Article  Google Scholar 

  • P. Walley (1991) Statistical Reasoning with Imprecise Probabilities Chapman and Hall London

    Google Scholar 

  • P. Walley (1996) ArticleTitleMeasures of uncertainty in expert systems Artificial Intelligence 83 1–58 Occurrence Handle10.1016/0004-3702(95)00009-7

    Article  Google Scholar 

  • P. Walley L. Gurrin P. Burton (1996) ArticleTitleAnalysis of clinical data using imprecise prior probabilities The Statistician 45 457–485

    Google Scholar 

  • Walley, P. Pelessoni, R. and Vicig, P. (1999), Direct algorithms for checking coherence and making inferences from confiditional probability assessments. Quad. n. 6/99 del Dipartimento di Matematica Applicata “Bruno de Finetti”, University of Trieste.

  • White III, C.C. (1986), A posteriori representations based on linear inequality descriptions of a priori and conditional distributions, IEEE Transactions on Systems, Man and Cybernetics, SMC-16, 570–573.

  • L.J. Wolfson J.B. Kadane M.J. Small (1996) ArticleTitleBayesian environmental policy decisions: two case studies Ecological Applications 6 1056–1066

    Google Scholar 

  • J.F. Yates L.G. Zukowski (1976) ArticleTitleCharacterization of ambiguity in decision making Behavioral Science 21 19–25

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moore School of Business, University of South Carolina, Columbia, SC, 29208, U.S.A

    Kathleen M. Whitcomb

Authors
  1. Kathleen M. Whitcomb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kathleen M. Whitcomb.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Whitcomb, K.M. Quasi-Bayesian Analysis Using Imprecise Probability Assessments And The Generalized Bayes’ Rule. Theor Decis 58, 209–238 (2005). https://doi.org/10.1007/s11238-005-2458-y

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11238-005-2458-y

Key words

Search

Navigation