A Bayesian classifier

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

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

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.