On the error probability of model selection for classification

We estimate a conditional probability P(y|x) of class y∈Y given attribute x∈X from training examples, where X and Y are respectively infinite and finite sets. The estimated conditional probability is used for classification in which a class y is guessed from an attribute x based on the conditional probability P(y|x). The procedure can be also applied to order identification of Markov models. We derive the asymptotically exact error probability in model selection for an arbitrary function d(·) which determines the selection procedure as well as the information criterion