  Feature selection aims to select the smallest subset of features for a
specified level of performance. The optimal achievable classification
performance on a feature subset is summarized by its Receiver Operating Curve
(ROC). When infinite data is available, the Neyman- Pearson (NP) design
procedure provides the most efficient way of obtaining this curve. In practice
the design procedure is applied to density estimates from finite data sets. We
perform a detailed statistical analysis of the resulting error propagation on
finite alphabets. We show that the estimated performance curve (EPC) produced
by the design procedure is arbitrarily accurate given sufficient data,
independent of the size of the feature set. However, the underlying likelihood
ranking procedure is highly sensitive to errors that reduces the probability
that the EPC is in fact the ROC. In the worst case, guaranteeing that the EPC
is equal to the ROC may require data sizes exponential in the size of the
feature set. These results imply that in theory the NP design approach may only
be valid for characterizing relatively small feature subsets, even when the
performance of any given classifier can be estimated very accurately. We
discuss the practical limitations for on-line methods that ensures that the NP
procedure operates in a statistically valid region.
