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 $X_1,X_2,\dots$ 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 $M_n \in {1,2,\ldots,S}$ateachtimeaccordingtotherule$M_n = f(M_{n-1},X_n)$, where $f$ is a deterministic time-invariant function. Assume that we let the process run for a very long time ($n\to \mathrm{\infty })$, 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 $P_e$ 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.