Abstract
Computational procedures using independence assumptions in various forms are popular in machine learning, although checks on empirical data have given inconclusive results about their impact. Some theoretical understanding of when they work is available, but a definite answer seems to be lacking. This paper derives distributions that maximizes the statewise difference to the respective product of marginals. These distributions are, in a sense the worst distribution for predicting an outcome of the data generating mechanism by independence. We also restrict the scope of new theoretical results by showing explicitly that, depending on context, independent (’Naïve’) classifiers can be as bad as tossing coins. Regardless of this, independence may beat the generating model in learning supervised classification and we explicitly provide one such scenario.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Russell, S., Norvig, P.: Artificial intelligence: a modern approach. Prentice-Hall, Englewood Cliffs (1995)
Chow, C., Liu, C.: Approximating discrete probability distributions with dependency trees. IEEE Transactions on Information Theory 14(3), 462–467 (1968)
Heckerman, D., Geiger, D., Chickering, D.: Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning Journal 20(3), 197–243 (1995)
Hand, D., Yu, K.: Idiot’s bayes–not so stupid after all? International Statistical Review 69(3), 385–398 (2001)
Lewis, P.: Approximating probability distributions to reduce storage requirements. Information and Control 2, 214–225 (1959)
Vapnik, V.: Statistical Learning Theory. Wiley, Chichester (1998)
Catoni, O.: Statistical Learning Theory and Stochastic Optimization. Springer, Heidelberg (2004)
Devroye, L., Györfi, L., Lugosi, G.: A Probabilistic Theory of Pattern Recognition. Springer, Heidelberg (1996)
Huang, K., King, I., Lyu, M.: Finite mixture model of bounded semi-naive Bayesian network classifier. In: Kaynak, O., Alpaydın, E., Oja, E., Xu, L. (eds.) ICANN 2003 and ICONIP 2003. LNCS, vol. 2714, Springer, Heidelberg (2003)
Ripley, B.: Pattern Recognition and Neural Networks. Cambridge University Press, Cambridge (1996)
Titterington, D., Murray, G., Murray, L., Spiegelhalter, D., Skene, A., Habbema, J., Gelpke, G.: Comparison of discrimination techniques applied to a complex data set of head injured patients. Journal of the Royal Statistical Society 144(2), 145–175 (1981)
Chickering, D.: Learning equivalence classes of bayesian-network structures. The Journal of Machine Learning Research 2, 445–498 (2002)
Rish, I., Hellerstein, J., Thathachar, J.: An analysis of data characteristics that affect naive bayes performance. Technical Report RC21993, IBM (2001)
Ekdahl, M.: Approximations of Bayes Classifiers for Statistical Learning of Clusters. Licentiate thesis, Linköpings Universitet (2006)
Ekdahl, M., Koski, T., Ohlson, M.: Concentrated or non-concentrated discrete distributions are almost independent. IEEE Transactions on Information Theory (submitted)
Domingos, P., Pazzani, M.: On the optimality of the simple bayesian classifier under zero-one loss. Machine Learning 29(2), 103–130 (1997)
Ekdahl, M., Koski, T.: Bounds for the loss in probability of correct classification under model based approximation. Journal of Machine Learning Research 7, 2473–2504 (2006)
Hagerup, T., Rub, C.: A guided tour of Chernoff bounds. Information Processing Letters 33, 305–308 (1989)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ekdahl, M., Koski, T. (2007). On Concentration of Discrete Distributions with Applications to Supervised Learning of Classifiers. In: Perner, P. (eds) Machine Learning and Data Mining in Pattern Recognition. MLDM 2007. Lecture Notes in Computer Science(), vol 4571. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73499-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-73499-4_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73498-7
Online ISBN: 978-3-540-73499-4
eBook Packages: Computer ScienceComputer Science (R0)