Skip to main content
SpringerLink
Log in

Uncertainty, credal sets and second order probability

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

The last 20 years or so has seen an intense search carried out within Dempster–Shafer theory, with the aim of finding a generalization of the Shannon entropy for belief functions. In that time, there has also been much progress made in credal set theory—another generalization of the traditional Bayesian epistemic representation—albeit not in this particular area. In credal set theory, sets of probability functions are utilized to represent the epistemic state of rational agents instead of the single probability function of traditional Bayesian theory. The Shannon entropy has been shown to uniquely capture certain highly intuitive properties of uncertainty, and can thus be considered a measure of that quantity. This article presents two measures developed with the purpose of generalizing the Shannon entropy for (1) unordered convex credal sets and (2) possibly non-convex credal sets ordered by second order probability, thereby providing uncertainty measures for such epistemic representations. There is also a comparison with the results of the measure AU developed within Dempster–Shafer theory in a few instances where unordered convex credal set theory and Dempster–Shafer theory overlap.

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

  • Abellán J. (2006) Uncertainty measures on probability intervals from the imprecise Dirichlet model. International Journal of General Systems 35(5): 509–528

    Article  Google Scholar 

  • Abellán J., Klir G. J., Moral S. (2006) Disaggregated total uncertainty measure for credal sets. International Journal of General Systems 35(1): 29–44

    Article  Google Scholar 

  • Couso I., Moral S., Walley P. (1999) Examples of independence for imprecise probabilities. ISIPA, Ghent

    Google Scholar 

  • Cozman F. G. (1999a) Calculation of posterior bounds given convex sets of prior probability measures and likelihood functions. Journal of Computational and Graphical Statistics 8(4): 824–838

    Google Scholar 

  • Cozman, F. (1999b). A brief introduction to the theory of sets of probability functions. Retrieved June 17, 2008 from http://www.poli.usp.br/p/fabio.cozman/Research/CredalSetsTutorial/postscript.html.

  • Ellsberg, D. (1961). Risk, ambiguity and the savage axioms. Quarterly Journal of Economics, 75, 643–669. (Reprinted in P., Gärdenfors & N. -E., Sahlin (Eds.) (1988). Decision, probability and utility. Cambridge: Cambridge University Press).

    Google Scholar 

  • Floridi, L. (2007). Semantic conceptions of information. In E. N. Zalta (Ed.), The stanford encyclopedia of philosophy. New York: Springer. Retrieved June 17, 2008, from http://plato.stanford.edu/archives/spr2007/entries/information-semantic/.

  • Gajdos T., Hayashi T., Tallon J.-C., Vergnaud J.-C. (2008) Attitude towards imprecise information. Journal of Economic Theory 140(1): 27–65

    Article  Google Scholar 

  • Gilboa I., Schmeidler D. (1989) Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18: 141–153

    Article  Google Scholar 

  • Gärdenfors, P., & Sahlin, N. -E. (1982). Unreliable probabilities, risk taking, and decision making. Synthese, 53, 361–386 (Reprinted in P., Gärdenfors & N. -E., Sahlin (1988). Decision, probability and utility. Cambridge: Cambridge University Press).

  • Harmanec, D. (1999). Measures of uncertainty and information. Retrieved June 17, 2008, from www.sipta.org/documentation.

  • Harmanec D., Klir G. J. (1994) Measuring total uncertainty in Dempster–Shafer theory: A novel approach. International Journal of General Systems 22: 405–419

    Article  Google Scholar 

  • Harms W. F. (1998) The use of information theory in epistemology. Philosophy of science 65(3): 472–501

    Article  Google Scholar 

  • Klir G. J. (1995) Principles of uncertainty: What are they? Why do we need them?. Fuzy Sets and Systems 74(1): 15–31

    Article  Google Scholar 

  • Klir, G. J. (1999). Uncertainty and information: Measures for imprecise probabilities: An overview. 1st International symposium on imprecise probabilities and their applications. Ghent: ISIPA. Retrieved June 17, 2008, from http://www.poli.usp.br/d/pmr5406/Download/papers/Klir_uncertainty-and-information-measures.ps.

  • Klir G.J., Folger T. A. (1988) Fuzzy sets, uncertainty and information. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Klir G. J., Smith R. M. (2001) On measuring uncertainty and uncertainty-based information: Recent developments. Annals of Mathematics and Artificial Intelligence 32: 5–33

    Article  Google Scholar 

  • Lewis D. (1986) Philosophical papers, Vol. 2. Oxford University Press, Oxford

    Google Scholar 

  • Lewis D. (1994) Humean supervenience debugged. Mind 103(412): 473–490

    Article  Google Scholar 

  • Levi I. (1974) On indeterminate probabilities. The Journal of Philosophy 71(13): 391–418

    Article  Google Scholar 

  • Levi I. (1980) The enterprise of knowledge. MIT Press, Cambridge

    Google Scholar 

  • Levi I. (1982) Ignorance, probability and rational choice. Synthese 53: 387–417

    Article  Google Scholar 

  • Rényi, A. (1961). On measures of entropy and information. Proceedings of the 4 th Berkeley symposium on mathematics, statistics and probability (pp. 547–561). California: University of California Press.

  • Savage L. J. (1972) The foundations of statistics. Dover Publications, New York

    Google Scholar 

  • Seidenfeld T. (1979) Why i am not an objective Bayesian; some reflections prompted by Rosenkrantz. Theory and Decision 11: 413–440

    Article  Google Scholar 

  • Seidenfeld T. (1986) Entropy and uncertainty. Philosophy of Science 53: 467–491

    Article  Google Scholar 

  • Shannon, C. E. (1948). A mathematical theory of communication [electronic version]. Bell System Technical Journal, 27, 379–423 and 623–656. Retrieved June 17, 2008 from http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html.

  • Smets, P. (1990). The transferable belief model and other interpretations of Dempster–Shafer’s model. In Proceedings of the sixth annual conference on uncertainty in artificial intelligence (pp. 375–384). Brussels: Université Libre de Bruxelles.

  • van Fraassen B. (1990) Figures in a probability landscape. In: Dunn J., Gupta A. (eds) Truth or consequences. Kluwer, Dordrecht

    Google Scholar 

  • Weatherson B. (2002) Keynes, uncertainty and interest rates. Cambridge Journal of Economics 26: 47–62

    Article  Google Scholar 

  • Zynda L. (2000) Representation theorems and realism about degrees of belief. Philosophy of Science 67: 45–69

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. KTH Royal Institute of Technology, Teknikringen, 78 B, Stockholm, Sweden

    Jonas Clausen Mork

Authors
  1. Jonas Clausen Mork
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonas Clausen Mork.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mork, J.C. Uncertainty, credal sets and second order probability. Synthese 190, 353–378 (2013). https://doi.org/10.1007/s11229-011-0042-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-011-0042-2

Keywords

Search

Navigation