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Theory of Probability and Mathematical Statistics

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A Bayesian classifier

Authors: B. A. Zalessky and P. V. Lukashevich
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 23-35
MSC (2000): Primary 62C10; Secondary 90Bxx
Published electronically: August 4, 2009
MathSciNet review: 2446846
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Abstract: We consider a new Bayesian classifier for the classification of multidimensional observations $ X_1,\dots,X_n$ of $ \mathbb{R}^k$ if the learning sample is known. We assume that the data are generated by two disjoint bounded sets $ \Omega_0,\Omega_1\subset \mathbb{R}^k$ and each vector $ X_i$ of the sample is a result of the observation after one of the sets $ \Omega_\ell$, $ \ell=0,1$, with a random error. In other words, we assume that a priori the Bayesian probability $ \mu$ is given on the set $ \Omega=\Omega_0\cup\Omega_1$ and that every vector of observations $ X_i$ has the density

$\displaystyle g_\ell(x)=q_\ell\int_{\Omega_\ell}f(x,y) \mu(dy),\qquad \ell=0,1, $

where the function $ f(x,y)$ is a probability density for all $ y\in \Omega$ and $ q_\ell^{-1}=\mu(\Omega_\ell)$.

The maximum a posteriori probability estimators $ \widehat{\Omega}_{\ell,n}$, $ \ell=0,1$, for the sets  $ \Omega_\ell$, $ \ell=0,1$, are constructed with the help of the learning sample. Under natural assumptions imposed on $ \Omega_0$ and $ \Omega_1$, we show that the estimators converge to some sets (possibly different from $ \Omega_0$ and $ \Omega_1$). If the mean frequencies $ \pi_\ell$ of observations of the classes $ \Omega_\ell$ are equal to $ \mu(\Omega_\ell)$, $ \ell=0,1$, then the estimators are consistent in the sense that $ \widehat{\Omega}_{\ell,n} \stackrel{n\rightarrow\infty}{\longrightarrow}\Omega_{\ell}$, $ \ell=0,1$. We also discuss some results of numerical experiments showing the applicability of our classifier for solving the problems of the statistical classification.

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Additional Information

B. A. Zalessky
Affiliation: United Institute of Informatics Problems, National Academy of Sciences, Surganova Street 6, Minsk, 220012, Belarus’

P. V. Lukashevich
Affiliation: United Institute of Informatics Problems, National Academy of Sciences, Surganova Street 6, Minsk, 220012, Belarus’

Received by editor(s): October 23, 2006
Published electronically: August 4, 2009
Additional Notes: The first author was supported by the INTAS grant 04-77-7036
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society