Assume that we are allowed to reject an input as unrecognizable in a pattern-classification task. For an input x belonging to class !, we can define a new loss function for any decision rule g¹xº as follows:

where r 2 ¹0, 1º is the loss incurred for choosing a rejection action. Derive the optimal decision rule for this three-way loss function.

I(w, g(x)) = m = (x)8 : m = (x)80 1 :rejection,