Learning with Finite Memory

Abstract

This paper develops the theory of the design and performance of optimal finite-memory systems for the two-hypothesis testing problem. Let X1, X2, ⋯ be a sequence of independent identically distributed random variables drawn according to a probability measure P. Consider the standard two-hypothesis testing problem with probability of error loss criterion in which P = P0 with probability π0; and P = P1 with probability π1. Let the data be summarized after each new observation by an m-valued statistic T∈{ 1, 2, ⋯, m} which is updated according to the rule Tn = f(Tn-1, Xn), where f is a (perhaps randomized) time-invariant function. Let d:{ 1, 2,⋯, m} →{ H0, H1} be a fixed decision function taking action d(Tn) at time n, and let Pe(f,d) be the long-run probability of error of the algorithm (f, d) as the number of trials n→∞. Define $P^\ast = \inf_{(f,d)}P_e(f, d)$ . Let the a.e. maximum and minimum likelihood ratios be defined by $\bar{l} = \sup(\mathscr{P}_0(A)/\mathscr{P}_1(A))$ and $\underline{l} = \inf(\mathscr{P}_0(A)/\mathscr{P}_1(A))$ where the supremum and infimum are taken over all measurable sets A for which $\mathscr{P}_0(A) + \mathscr{P}_1(A) > 0$ . Define $\gamma = \bar{l}/\underline{l}$ . It will be shown that P* = [ 2(π0π1γm-1)1/2 - 1]/(γm-1 - 1), under the nondegeneracy condition γm-1 ≥ max{π0/π1, π1/π0}; and a simple family of ε-optimal (f, d)'s will be exhibited. In general, an optimal (f, d) does not exist; and ε-optimal algorithms involve randomization in f.