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 $X1,X2,\ldots$ 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 $Mn \in {1,2,\ldots,S}$ at each time according to the rule $Mn = f(M{n-1},Xn)$, where $f$ is a deterministic time-invariant function. Assume that we let the process run for a very long time ($n\rightarrow \infty)$, 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.