Abstract
From n categorical observations, what can be predicted about the next n′ ones? We present a generalization of the Bayesian approach, the imprecise Dirichlet-multinomial model (IDMM), which uses sets of Dirichlet-multinomial distributions to model prior ignorance. The IDMM satisfies coherence, symmetry and several desirable invariance properties.
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References
Bernard, J.-M.: Bayesian interpretation of frequentist procedures for a Bernoulli process. Amer. Statist. 50, 7–13 (1996)
Bernard, J.-M.: Bayesian inference for categorized data. In: Rouanet, H., Bernard, J.-M., Bert, M.-C., Lecoutre, B., Lecoutre, M.-P., Le Roux, B. (eds.) New Ways in Statistical Methodology: From Significance Tests to Bayesian Inference. European University Studies, Psychology. Peter Lang, Bern., vol. 6, pp. 159–226 (1998)
Bernard, J.-M.: An introduction to the imprecise Dirichlet model for multinomial data. Internat. J. Approx. Reason 39, 123–150 (2005)
Cox, D.R., Hinkley, D.V.: Theoretical Statistics. Chapman and Hall, London (1974)
De Cooman, G., Miranda, E., Quaeghebeur, E.: Representation insensitivity in immediate prediction under exchangeability. Internat. J. Approx. Reason (in press, 2008)
De Finetti, B.: Theory of Probability, vol. 1. John Wiley & Sons, Chichester (1974)
Geisser, S.: On prior distributions for binary trials (with discussion). Amer. Statist. 38, 244–251 (1984)
Geisser, S.: Predictive inference: An introduction. In: Monographs on Statistics and Applied Probability, vol. 55, Chapman and Hall, New York (1993)
Johnson, N.L., Kotz, S., Balakrishnan, N.: Discrete Multivariate Distributions. Wiley, New York (1997)
Perks, W.: Some observations on inverse probability including a new indifference rule (with discussion). J. Inst. Actuar. 73, 285–334 (1947)
Thatcher, A.R.: Relationships between Bayesian and confidence limits for predictions (with discussion). J. Roy. Statist. Soc. Ser. B 26, 176–210 (1964)
Walley, P.: Statistical reasoning with imprecise probabilities. In: Monographs on Statistics and Applied Probability, vol. 42. Chapman and Hall, London (1991)
Walley, P.: Inferences from multinomial data: learning about a bag of marbles. J. Roy. Statist. Soc. Ser. B 58, 3–57 (1996)
Walley, P.: Reconciling frequentist properties with the likelihood principle. J. Statist. Plann. Inference 105, 35–65 (2002)
Walley, P., Bernard, J.-M.: Imprecise probabilistic prediction for categorical data. Technical Report CAF-9901, Laboratoire Cognition et Activités finalisées, Université Paris 8, Saint-Denis, France (1999)
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Bernard, JM. (2008). Imprecise Probabilistic Prediction for Categorical Data: From Bayesian Inference to the Imprecise Dirichlet-Multinomial Model. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_1
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DOI: https://doi.org/10.1007/978-3-540-85027-4_1
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