Abstract
We propose two models, one continuous and one categorical, to learn about dependence between two random variables, given only limited joint observations, but assuming that the marginals are precisely known. The continuous model focuses on the Gaussian case, while the categorical model is generic. We illustrate the resulting statistical inferences on a simple example concerning the body mass index. Both methods can be extended easily to three or more random variables.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Augustin, T., Coolen, F.P.A., de Cooman, G., Troffaes, M.C.M.: Introduction to imprecise probabilities. Edited book (submitted to publisher)
Bernado, J.M., Smith, A.F.M.: Bayesian Theory. John Wiley and Sons (1994)
Boole, G.: An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities. Walton and Maberly, London (1854)
Ferson, S., Kreinovich, V., Ginzburg, L., Myers, D.S., Sentz, K.: Constructing probability boxes and Dempster-Shafer structures. Technical Report SAND2002–4015, Sandia National Laboratories (January 2003)
Hand, D.J., Daly, F., McConway, K., Lunn, D., Ostrowski, E.: A handbook of small data sets. CRC Press (1993)
Kass, R.E., Wasserman, L.: The selection of prior distributions by formal rules. Journal of the American Statistical Association 91(435), 1343–1370 (1996)
Nelsen, R.B.: An introduction to copulas. Springer (1999)
Quaeghebeur, E., de Cooman, G.: Imprecise probability models for inference in exponential families. In: Cozman, F.G., Nau, R., Seidenfeld, T. (eds.) ISIPTA 2005: Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, Pittsburgh, USA, pp. 287–296 (July 2005)
Pelessoni, R., Vicig, P., Montes, I., Miranda, E.: Imprecise copulas and bivariate stochastic orders. In: De Baets, B., Fodor, J., Montes, S. (eds.) Proceedings of Eurofuse 2013 Workshop, pp. 217–225 (2013)
Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959)
Troffaes, M.C.M., Blake, S.: A robust data driven approach to quantifying common-cause failure in power networks. In: Cozman, F., Denœux, T., Destercke, S., Seidenfeld, T. (eds.) ISIPTA 2013: Proceedings of the Eighth International Symposium on Imprecise Probability: Theories and Applications, Compiègne, France, pp. 311–317. SIPTA (July 2013)
Troffaes, M.C.M., Destercke, S.: Probability boxes on totally preordered spaces for multivariate modelling. International Journal of Approximate Reasoning 52(6), 767–791 (2011)
Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)
Walley, P.: Inferences from multinomial data: Learning about a bag of marbles. Journal of the Royal Statistical Society, Series B 58(1), 3–34 (1996)
Walter, G., Augustin, T.: Imprecision and prior-data conflict in generalized Bayesian inference. Journal of Statistical Theory and Practice 3, 255–271 (2009)
Williamson, R.C., Downs, T.: Probabilistic arithmetic I: Numerical methods for calculating convolutions and dependency bounds. International Journal of Approximate Reasoning 4, 89–158 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Troffaes, M.C.M., Coolen, F.P.A., Destercke, S. (2014). A Note on Learning Dependence under Severe Uncertainty. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2014. Communications in Computer and Information Science, vol 444. Springer, Cham. https://doi.org/10.1007/978-3-319-08852-5_51
Download citation
DOI: https://doi.org/10.1007/978-3-319-08852-5_51
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08851-8
Online ISBN: 978-3-319-08852-5
eBook Packages: Computer ScienceComputer Science (R0)