On Strong Consistency of Model Selection in Classification
Joe Suzuki · IEEE Transactions on Information Theory · 2006
This paper considers model selection in classification. In many applications such as pattern recognition, probabilistic inference using a Bayesian network, prediction of the next in a sequence based on a Markov chain, the conditional probability P(Y=y|X=x) of class yisinY given attribute value xisinX is utilized. By model we mean the equivalence relation in X: for x,x'isinXx~x'hArrP(Y=y|X=x)=P(Y=y|X=x'), forall yisinY. By classification we mean the number of such equivalence classes is finite. We estimate the model from n samples zn=(xi,yi)i=1nisin(XtimesY)n, using information criteria in the form empirical entropy H plus penalty term (k/2)dn(the model such that H+(k/2)dnis minimized is the estimated model), where k is the number of independent parameters in the model, and {dn}n=1infinis a real nonnegative sequence such that lim supndn/n=0. For autoregressive processes, although the definitions of H and k are different, it is known that the estimated model almost surely coincides with the true model as nrarrinfin if {dn}n=1infin>{2loglogn}n=1infin, and that it does not if {dn}n=1infinn=1infin(Hannan and Quinn). The problem whether the same property is true for classification was open. This paper solves the problem in the affirmative