Binary Hypothesis Testing with Deterministic Finite-Memory Decision Rules
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In this paper we consider the problem of binary hypothesis testing with finite memory systems. Let X_1,X_2,... be a sequence of independent identically distributed Bernoulli random variables, with expectation p under H_0 and q under H_1. Consider a finite-memory deterministic machine with S states that updates its state M_n ∈{1,2,...,S} at each time according to the rule M_n = f(M_n-1,X_n), where f is a deterministic time-invariant function. Assume that we let the process run for a very long time (n→∞), and then make our decision according to some mapping from the state space to the hypothesis space. The main contribution of this paper is a lower bound on the Bayes error probability P_e of any such machine. In particular, our findings show that the ratio between the maximal exponential decay rate of P_e with S for a deterministic machine and for a randomized one, can become unbounded, complementing a result by Hellman.
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