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Dempster-Shafer theory and statistical inference with weak beliefs. (English) Zbl 1328.62040

Summary: 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.

MSC:

62A01 Foundations and philosophical topics in statistics
62G10 Nonparametric hypothesis testing
68T30 Knowledge representation
68T37 Reasoning under uncertainty in the context of artificial intelligence
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References:

[1] Dempster, A. P. (1963). Further examples of inconsistencies in the fiducial argument. Ann. Math. Statist. 34 884-891. · Zbl 0214.17902 · doi:10.1214/aoms/1177704011
[2] Dempster, A. P. (1966). New methods for reasoning towards posterior distributions based on sample data. Ann. Math. Statist. 37 355-374. · Zbl 0178.54302 · doi:10.1214/aoms/1177699517
[3] Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38 325-339. · Zbl 0168.17501 · doi:10.1214/aoms/1177698950
[4] Dempster, A. P. (1968). A generalization of Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 30 205-247. JSTOR: · Zbl 0169.21301
[5] Dempster, A. P. (1969). Upper and lower probability inferences for families of hypotheses with monotone density ratios. Ann. Math. Statist. 40 953-969. · Zbl 0211.50204 · doi:10.1214/aoms/1177697600
[6] Dempster, A. P. (2008). Dempster-Shafer calculus for statisticians. Internat. J. Approx. Reason. 48 265-277. · Zbl 1274.62053 · doi:10.1016/j.ijar.2007.03.004
[7] Denoeux, T. (2006). Constructing belief functions from sample data using multinomial confidence regions. Internat. J. Approx. Reason. 42 228-252. · Zbl 1100.68112 · doi:10.1016/j.ijar.2006.01.001
[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. · Zbl 1166.62004 · doi:10.1214/00-AOAS223
[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. · Zbl 0011.03205
[11] Fraser, D. A. S. (1968). The Structure of Inference . Wiley, New York. · Zbl 0164.48703
[12] Hannig, J. (2009). On generalized fiducial inference. Statist. Sinica 19 491-544. · Zbl 1168.62004 · doi:10.1007/s10114-008-6010-1
[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. · Zbl 1239.62008 · doi:10.1016/j.ijar.2007.05.003
[14] Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications , 2nd ed. Springer, New York. · Zbl 1026.62084
[15] Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. Roy. Statist. Soc. Ser. B 20 102-107. JSTOR: · Zbl 0085.35503
[16] Martin, R. and Ghosh, J. K. (2008). Stochastic approximation and Newton’s estimate of a mixing distribution. Statist. Sci. 23 365-382. · Zbl 1329.62361 · doi:10.1214/08-STS265
[17] Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statist. 22 400-407. · Zbl 0054.05901 · doi:10.1214/aoms/1177729586
[18] Shafer, G. (1976). A Mathematical Theory of Evidence . Princeton Univ. Press, Princeton, NJ. · Zbl 0359.62002
[19] Shafer, G. (1978/79). Nonadditive probabilities in the work of Bernoulli and Lambert. Arch. Hist. Exact Sci. 19 309-370. · Zbl 0392.01010 · doi:10.1007/BF00330065
[20] Shafer, G. (1979). Allocations of probability. Ann. Probab. 7 827-839. · Zbl 0414.60002 · doi:10.1214/aop/1176994941
[21] Shafer, G. (1981). Constructive probability. Synthese 48 1-60. · Zbl 0522.60001 · doi:10.1007/BF01064627
[22] Shafer, G. (1982). Belief functions and parametric models (with discussion). J. Roy. Statist. Soc. Ser. B 44 322-352. JSTOR: · Zbl 0499.62007
[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. · Zbl 1135.68051 · doi:10.1007/978-3-540-44792-4
[24] Zabell, S. L. (1992). R. A. Fisher and the fiducial argument. Statist. Sci. 7 369-387. · Zbl 0955.62521 · doi:10.1214/ss/1177011233
[25] Zhang, J. and Liu, C. (2010). Dempster-Shafer inference with weak beliefs. Statistica Sinica . · Zbl 1286.62015 · doi:10.5705/ss.2011.022a
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