Properties and convergence of a posteriori probabilities in classification problems

This report investigates the behavior of the a posteriori probabilities for classification problems in which the observations are not identically distributed. Some basic properties of the a posteriori probabilities are presented; then, it is shown that for each class the a posteriori probability converges a.s. to a random variable. Conditions are given for a.s. convergence of the a posteriori probability to 1 for the true class (and to 0 for all other classes). The results are illustrated for the case of two classes and binary observations, and finally a numerical example is presented.