Statistical Science

Dempster–Shafer Theory and Statistical Inference with Weak Beliefs

Ryan Martin, Jianchun Zhang, and Chuanhai Liu

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Abstract

The Dempster–Shafer (DS) theory is a powerful tool for probabilistic reasoning based on a formal calculus for combining evidence. DS theory has been widely used in computer science and engineering applications, but has yet to reach the statistical mainstream, perhaps because the DS belief functions do not satisfy long-run frequency properties. Recently, two of the authors proposed an extension of DS, called the weak belief (WB) approach, that can incorporate desirable frequency properties into the DS framework by systematically enlarging the focal elements. The present paper reviews and extends this WB approach. We present a general description of WB in the context of inferential models, its interplay with the DS calculus, and the maximal belief solution. New applications of the WB method in two high-dimensional hypothesis testing problems are given. Simulations show that the WB procedures, suitably calibrated, perform well compared to popular classical methods. Most importantly, the WB approach combines the probabilistic reasoning of DS with the desirable frequency properties of classical statistics.

Article information

Source
Statist. Sci. Volume 25, Number 1 (2010), 72-87.

Dates
First available in Project Euclid: 3 August 2010

Permanent link to this document
http://projecteuclid.org/euclid.ss/1280841734

Digital Object Identifier
doi:10.1214/10-STS322

Mathematical Reviews number (MathSciNet)
MR2741815

Zentralblatt MATH identifier
05945286

Citation

Martin, Ryan; Zhang, Jianchun; Liu, Chuanhai. Dempster–Shafer Theory and Statistical Inference with Weak Beliefs. Statist. Sci. 25 (2010), no. 1, 72--87. doi:10.1214/10-STS322. http://projecteuclid.org/euclid.ss/1280841734.


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References

  • [1] Dempster, A. P. (1963). Further examples of inconsistencies in the fiducial argument. Ann. Math. Statist. 34 884–891.
  • [2] Dempster, A. P. (1966). New methods for reasoning towards posterior distributions based on sample data. Ann. Math. Statist. 37 355–374.
  • [3] Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38 325–339.
  • [4] Dempster, A. P. (1968). A generalization of Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 30 205–247.
  • [5] Dempster, A. P. (1969). Upper and lower probability inferences for families of hypotheses with monotone density ratios. Ann. Math. Statist. 40 953–969.
  • [6] Dempster, A. P. (2008). Dempster–Shafer calculus for statisticians. Internat. J. Approx. Reason. 48 265–277.
  • [7] Denoeux, T. (2006). Constructing belief functions from sample data using multinomial confidence regions. Internat. J. Approx. Reason. 42 228–252.
  • [8] Edlefsen, P. T., Liu, C. and Dempster, A. P. (2009). Estimating limits from Poisson counting data using Dempster–Shafer analysis. Ann. Appl. Statist. 3 764–790.
  • [9] Fisher, R. A. (1930). Inverse probability. Proceedings of the Cambridge Philosophical Society 26 528–535.
  • [10] Fisher, R. A. (1935). The logic of inductive inference. J. Roy. Statist. Soc. 98 39–82.
  • [11] Fraser, D. A. S. (1968). The Structure of Inference. Wiley, New York.
  • [12] Hannig, J. (2009). On generalized fiducial inference. Statist. Sinica 19 491–544.
  • [13] Kohlas, J. and Monney, P.-A. (2008). An algebraic theory for statistical information based on the theory of hints. Internat. J. Approx. Reason. 48 378–398.
  • [14] Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Springer, New York.
  • [15] Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. Roy. Statist. Soc. Ser. B 20 102–107.
  • [16] Martin, R. and Ghosh, J. K. (2008). Stochastic approximation and Newton’s estimate of a mixing distribution. Statist. Sci. 23 365–382.
  • [17] Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22 400–407.
  • [18] Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton Univ. Press, Princeton, NJ.
  • [19] Shafer, G. (1978/79). Nonadditive probabilities in the work of Bernoulli and Lambert. Arch. Hist. Exact Sci. 19 309–370.
  • [20] Shafer, G. (1979). Allocations of probability. Ann. Probab. 7 827–839.
  • [21] Shafer, G. (1981). Constructive probability. Synthese 48 1–60.
  • [22] Shafer, G. (1982). Belief functions and parametric models (with discussion). J. Roy. Statist. Soc. Ser. B 44 322–352.
  • [23] Yager, R. and Liu, L. (eds.) (2008). Classic Works of the Dempster–Shafer Theory of Belief Functions. Stud. Fuzziness Soft Comput. 219. Springer, Berlin.
  • [24] Zabell, S. L. (1992). R. A. Fisher and the fiducial argument. Statist. Sci. 7 369–387.
  • [25] Zhang, J. and Liu, C. (2010). Dempster–Shafer inference with weak beliefs. Statistica Sinica. To appear.