A Bayesian classifier
Boris Zalessky, Pavel Lukashevich · Theory of Probability and Mathematical Statistics · 2009
We consider a new Bayesian classifier for the classification of multidimensional observations X 1 , . . ., X n of R k if the learning sample is known.We assume that the data are generated by two disjoint bounded sets Ω 0 , Ω 1 ⊂ R k and each vector X i of the sample is a result of the observation after one of the sets Ω ℓ , ℓ = 0, 1, with a random error.In other words, we assume that a priori the Bayesian probability µ is given on the set Ω = Ω 0 ∪ Ω 1 and that every vector of observations X i has the densitywhere the function f (x, y) is a probability density for all y ∈ Ω and q -1 ℓ = µ(Ω ℓ ).The maximum a posteriori probability estimators Ω ℓ,n , ℓ = 0, 1, for the sets Ω ℓ , ℓ = 0, 1, are constructed with the help of the learning sample.Under natural assumptions imposed on Ω 0 and Ω 1 , we show that the estimators converge to some sets (possibly different from Ω 0 and Ω 1 ).If the mean frequencies π ℓ of observations of the classes Ω ℓ are equal to µ(Ω ℓ ), ℓ = 0, 1, then the estimators are consistent in the sense that Ω ℓ,n n→∞ -→ Ω ℓ , ℓ = 0, 1.We also discuss some results of numerical experiments showing the applicability of our classifier for solving the problems of the statistical classification.