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 X1, X2, …. 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 Mn ∈ {1,2,…, S} at each time according to the rule Mn = f(Mn-1, Xn), 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 Pe of any such machine. In particular, our findings show that the ratio between the maximal exponential decay rate of Pe with S for a deterministic machine and for a randomized one, can become unbounded, complementing a result by Hellman.