Non-parametric search for significant inputs of unknown system

Abstract

Suppose that we can assign arbitrary t-tuple of binary inputs x:=(x(i),i/spl isin/[t]),[t]:={1,...,t} and measure the output Z which depends only on unknown s-tuple x(A) of significant inputs (SIs). A priori, A is equally likely to be any s-tuple of different elements from [t]. The successive measurements Z/sub i/,i=1,...,N, are independent random variables (RVs) given an N-sequence (N-design) of input t-tuples (memoryless channel). For a sequence D of N designs, N=1,..., we introduce the asymptotic rate AR/sub T/(D)=lim sup/sub t/spl rarr//spl infin//log t/N(/spl gamma/) of a test T. Here N(/spl gamma/) is the minimal sample (block) size such that the probability of T to miss s-tuple A (mean error probability) is less than /spl gamma/,0</spl gamma/<1. It is shown for a general class of tests and designs that AR does not depend on /spl gamma/,0</spl gamma/<1, and remains the same for slowly decreasing /spl gamma/(t):|log /spl gamma/(t)|=o(log t) as t/spl rarr//spl infin/. This definition is well-applicable to the combinatorial problems (when noise is absent) whereas in the case of sequential design N(/spl gamma/) must be replaced with the mean duration of sequential strategy guaranteeing the same upper bound for the error probability. It is well-known that the maximum likelihood (ML)-decision minimizes the error probability for any design. This implies that AR/sub ML/ is optimal among all tests if applicable. We outline the progress in three directions.