A paradigm for class identification problems

The following problem arises in many applications involving classification, identification, and inference. There is a set of objects X, and a particular x ∈ X is chosen (unknown to us). Based on information obtained about x in a sequential manner, one wishes to decide whether x belongs to one class of objects A₀ or a different class of objects A₁. The authors study a general paradigm applicable to a broad range of problems of this type, which they refer to as problems of class identification or discernibility. They consider various types of information sequences, and various success criteria including discernibility in the limit, discernibility with a stopping criterion, uniform discernibility, and discernibility in the Cesaro sense. They consider decision rules both with and without memory. Necessary and sufficient conditions for discernibility are provided for each case in terms of separability conditions on the sets A ₀ and A₁. They then show that for any sets A₀ and A₁, various types of separability can be achieved by allowing failure on appropriate sets of small measure. Applications to problems in language identification, system identification, and discrete geometry are discussed