Abstract
This article proposes some related issues to classification problem by Bayesian method for two populations. They are relationships between Bayes error (BE) and other measures and the results for determining the BE. In addition, we propose three methods to find the prior probabilities that can make to reduce BE. The calculation of these methods can be performed conveniently and efficiently by the MATLAB procedures. The new approaches are tested by the numerical examples including synthetic and benchmark data and applied in medicine and economics. These examples also show the advantages of the proposed methods in comparison with existing methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer, Berlin (1985)
Devijver, P.A., Kittler, J.: Pattern Recognition: A Statistical Approach. Prentice Hall, New York (1982)
Dunn, J.C.: A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters. Cybern 3(3), 32–57 (2008)
Ghosh, A.K., Chaudhuri, P., Sengupta, D.: Classification using kernel density estimates. Technometrics 48(1), 377–392 (2012)
Inman, H.F., Bradley, E.L.: The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities. Commun. Stat. Theory Methods 18(10), 3851–3874 (1989)
James, I.: Estimation of the mixing proportion in a mixture of two normal distributions from simple, rapid measurements. Biometrics 5, 265–275 (1978)
Jasra, A., Holmes, C., Stephens, D.: Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Stat. Sci. 12, 50–67 (2005)
Kraft, C.H.: Some conditions for consistency and uniform consistency of statistical procedures. University of California (1955)
Martinez, W.L., Martinez, A.R.: Computational Statistics Handbook with MATLAB. CRC Press, Boca Raton (2007)
Matusita, K.: On the notion of affinity of several distributions and some of its applications. Ann. Inst. Stat. Math. 19(1), 181–192 (1967)
McLachlan, G.J., Basford, K.E.: Mixture Models. Inference and Applications to Clustering. Dekker, New York (1988)
Miller, G., Inkret, W., Little, T., Martz, H., Schillaci, M.: Bayesian prior probability distributions for internal dosimetry. Radiat. Protect. Dosim. 94(4), 347–352 (2001)
Nguyentrang, T., Vovan, T.: A new approach for determining the prior probabilities in the classification problem by Bayesian method. Adv. Data Anal. Classif. 11(3), 629–643 (2017)
Nielsen, F.: Generalized Bhattacharyya and Chernoff upper bounds on Bayes error using quasi-arithmetic means. Pattern Recognit. Lett. 42, 25–34 (2014)
Pal, N.R., Bezdek, J.C.: On cluster validity for the fuzzy c-means model. IEEE Trans. Fuzzy Syst. 3(3), 370–379 (1995)
Pham-Gia, T., Turkkan, N., Bekker, A.: Bounds for the bayes error in classification: a bayesian approach using discriminant analysis. Stat. Methods Appl. 16(1), 7–26 (2007)
Pham-Gia, T., Turkkan, N., Vovan, T.: Statistical discrimination analysis using the maximum function. Commun. Stat. Simul. Comput. 37(2), 320–336 (2008)
Scott, D.R.: Multivariate Density Estimation: Theory Practice and Visualization. Wiley, New York (1992)
Silverman, B.W.: Density Estimation for Statistics and Data Analysis. CRC Press, Boca Raton (1986)
Toussaint, G.: Some inequalities between distance measures for feature evaluation. Comput 21, 389–394 (1972)
Vovan, T.: \(L^1\)-distance and classification problem by Bayesian method. J. Appl. Stat. 44(3), 385–401 (2017)
Vovan, T., Nguyentrang, T.: Fuzzy clustering of probability density functions. J. Appl. Stat. 44(4), 583–601 (2017)
VoVan, T., Pham-Gia, T.: Clustering probability distributions. J. Appl. Stat. 37(11), 1891–1910 (2010)
Webb, A.R.: Statistical Pattern Recognition, 2nd edn. Wiley, Hoboken (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vovan, T., Tranphuoc, L. & Chengoc, H. Classifying Two Populations by Bayesian Method and Applications. Commun. Math. Stat. 7, 141–161 (2019). https://doi.org/10.1007/s40304-018-0139-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40304-018-0139-8