On Strong Consistency of Model Selection in Classification

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 z
ⁿ
=(x
_i
,y
_i
)
_i=1

ⁿ
isin(XtimesY)
ⁿ
, using information criteria in the form empirical entropy H plus penalty term (k/2)d
_n
(the model such that H+(k/2)d
_n
is minimized is the estimated model), where k is the number of independent parameters in the model, and {d
_n
}
_n=1

^infin
is a real nonnegative sequence such that lim sup
_n
d
_n
/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 {d
_n
}
_n=1

^infin
>{2loglogn}
_n=1

^infin
, and that it does not if {d
_n
}
_n=1

^infin
<{2loglogn}
_n=1

^infin
(Hannan and Quinn). The problem whether the same property is true for classification was open. This paper solves the problem in the affirmative