Binary Hypothesis Testing with Deterministic Finite-Memory Decision Rules

Abstract

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 H0 and q under H1. 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 timeinvariant 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.P e (f, d),