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.