# zbMATH — the first resource for mathematics

An introduction to the imprecise Dirichlet model for multinomial data. (English) Zbl 1066.62003
Summary: The imprecise Dirichlet model (IDM) was recently proposed by P. Walley [J. R. Stat. Soc., Ser. B 58, 3–57 (1996; Zbl 0834.62004); Statistical reasoning with imprecise probabilities (1991; Zbl 0732.62004)] as a model for objective statistical inference from multinomial data with chances $$\theta$$. In the IDM, prior or posterior uncertainty about $$\theta$$ is described by a set of Dirichlet distributions, and inferences about events are summarized by lower and upper probabilities. The IDM avoids shortcomings of alternative objective models, either frequentist or Bayesian. We review the properties of the model, for both parametric and predictive inferences, and some of its recent applications to various statistical problems.

##### MSC:
 62A01 Foundations and philosophical topics in statistics 62F15 Bayesian inference
Full Text:
##### References:
 [1] Abellán, J.; Moral, S., Building classification trees using the total uncertainty criterion, Int. J. intelligent syst., 18, 1215-1225, (2003) · Zbl 1101.68799 [2] Altham, P.M.E., Exact Bayesian analysis of a 2×2 contingency table and fisher’s exact significance test, J. roy. statist. soc. ser. B, 31, 2, 261-269, (1968) [3] Bernard, J.-M., Bayesian interpretation of frequentist procedures for a Bernoulli process, Amer. statist., 50, 7-13, (1996) [4] Bernard, J.-M., Bayesian analysis of tree-structured categorized data, Rev. int. systémique, 11, 1, 11-29, (1997) · Zbl 0891.62014 [5] J.-M. Bernard, Bayesian inference for categorized data, in: H. Rouanet (Ed.), New Ways in Statistical Methodology: From Significance Tests to Bayesian Inference, European University Studies, Series 6: Psychology, Peter Lang, Bern, 1998, pp. 159-226 [6] J.-M. Bernard, Non-parametric inference about an unknown mean using the imprecise Dirichlet model, in: G. de Cooman, T. Fine, T. Seidenfeld (Eds.), Proceedings of the 2nd International Symposium on Imprecise Probabilities and their Applications (ISIPTA’01), Shaker, Ithaca, New York, USA, 2001, pp. 40-50 [7] Bernard, J.-M., Implicative analysis for multivariate binary data using an imprecise Dirichlet model, J. statist. plann. inference, 105, 83-103, (2002) · Zbl 1007.62053 [8] J.-M. Bernard, Analysis of local or asymmetric dependencies in contingency tables using the imprecise Dirichlet model, in: J.-M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proceedings of the 3rd International Symposium on Imprecise Probabilities and their Applications (ISIPTA’03), Proceedings in Informatics, Carleton Scientific, Waterloo, Ontario, 18 (2003) 46-61 [9] Bernardo, J.M.; Smith, A.F.M., Bayesian theory, (1994), John Wiley New York [10] Bloch, D.A.; Watson, G.S., A Bayesian study of the multinomial distribution, Ann. math. statist., 38, 1423-1435, (1967) · Zbl 0183.21503 [11] Connor, R.J.; Mosimann, J.E., Concepts of independence for proportions with a generalization of the Dirichlet distribution, J. amer. statist. assoc., 64, 194-206, (1969) · Zbl 0179.24101 [12] Coolen, F., An imprecise Dirichlet model for Bayesian analysis of failure data including right-censored observations, Reliability engineering and system safety, 56, 61-68, (1997) [13] Cox, D.R.; Hinkley, D.V., Theoretical statistics, (1974), Chapman & Hall London · Zbl 0334.62003 [14] Fang, K.-T.; Kotz, S.; Ng, K.-W., Symmetric multivariate and related distributions, () [15] de Finetti, B., Theory of probability, (1974), John Wiley Chichester, vol. 1 [16] Geisser, S., Predictive inference: an introduction, () · Zbl 0824.62001 [17] Goodman, L.A., A single general method for the analysis of cross-classified data, J. amer. statist. assoc., 91, 408-428, (1996) · Zbl 0871.62051 [18] Hildebrand, D.K.; Laing, J.D.; Rosenthal, H., Prediction analysis of cross classifications, (1977), John Wiley New York · Zbl 0425.62005 [19] M. Hutter, Robust estimators under the imprecise Dirichlet model, in: J.-M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proceedings of the 3rd International Symposium on Imprecise Probabilities and their Applications (ISIPTA’03), Proceedings in Informatics, Carleton Scientific, Waterloo, Ontario, Canada 18 (2003) 274-289 [20] Jeffreys, H., Theory of probability, (1961), Oxford University Press Oxford · Zbl 0116.34904 [21] Kass, R.E.; Wasserman, L., The selection of prior distributions by formal rules, J. amer. statist. assoc., 91, 435, 1343-1370, (1996) · Zbl 0884.62007 [22] Kotz, S.; Balakrishnan, N.; Johnson, N.L., Continuous multivariate distributions. models and applications, (2000), John Wiley New York, vol. 1 · Zbl 0946.62001 [23] Pearson, K., The fundamental problem of practical statistics, Biometrika, 13, 1-16, (1920) [24] Perks, W., Some observations on inverse probability including a new indifference rule (with discussion), J. inst. actuaries, 73, 285-334, (1947) [25] E. Quaeghebeur, G. de Cooman, Game-theoretic learning using the imprecise Dirichlet model, in: J.-M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.), Proceedings of the 3rd International Symposium on Imprecise Probabilities and their Applications (ISIPTA’03), Proceedings in Informatics, Carleton Scientific, Waterloo, Ontario, Canada 18 (2003) 450-464 [26] Thatcher, A.R., Relationships between Bayesian and confidence limits for predictions (with discussion), J. roy. statist. soc. ser. B, 26, 176-210, (1964) · Zbl 0126.34403 [27] Walley, P., Statistical reasoning with imprecise probabilities, () · Zbl 0732.62004 [28] Walley, P., Inferences from multinomial data: learning about a bag of marbles, J. roy. statist. soc. ser. B, 58, 3-57, (1996) · Zbl 0834.62004 [29] Walley, P.; Gurrin, L.; Burton, P., Analysis of clinical data using imprecise prior probabilities, The Statistician, 45, 457-485, (1996) [30] P. Walley, J.-M. Bernard, Imprecise probabilistic prediction for categorical data, Technical Report CAF-9901, Laboratoire Cognition et Activités Finalisées, Université Paris 8, Saint-Denis, France, 1999 [31] Walley, P., Reconciling frequentist properties with the likelihood principle, J. statist. plann. inference, 105, 35-65, (2002) · Zbl 0992.62002 [32] Wilks, S.S., Mathematical statistics, (1962), John Wiley New York · Zbl 0173.45805 [33] Zabell, S.L., The rule of succession, Erkenntnis, 31, 283-321, (1989) [34] M. Zaffalon, Statistical inference of the naive credal classifier, in: G. de Cooman, T. Fine, S. Moral, T. Seidenfeld (Eds.), Proceedings of the 2nd International Symposium on Imprecise Probabilities and their Applications (ISIPTA’01), Shaker, Ithaca, New York, USA, 2001, pp. 384-393 [35] M. Zaffalon, Robust discovery of tree-dependency structures, in: G. de Cooman, T. Fine, S. Moral, T. Seidenfeld (Eds.), Proceedings of the 2nd International Symposium on Imprecise Probabilities and their Applications (ISIPTA’01), Shaker, Ithaca, New York, USA, 2001, pp. 394-403 [36] Zaffalon, M., The naive credal classifier, J. statist. plann. inference, 105, 1, 5-21, (2002) · Zbl 0992.62057 [37] M. Zaffalon, M. Hutter, Robust inference of trees, Technical Report IDSIA-11-03, IDSIA, Manno (Lugano), Switzerland, 2003 · Zbl 1177.68217
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.