Binary Hypothesis Testing with Deterministic Finite-Memory Decision Rules

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 {{\mathcal{H}}-0} and q under {{\mathcal{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.