Deterministic Finite-Memory Bias Estimation

TL;DR

This paper establishes the fundamental limit on the accuracy of finite-memory deterministic machines estimating a Bernoulli parameter, showing the minimax risk scales as 1/S, thus resolving a longstanding conjecture.

Contribution

It provides a tight upper bound matching the known lower bound, proving the minimax risk for S-state machines is Θ(1/S), and disproves the previous conjecture of Θ(log S / S).

Findings

Minimax asymptotic risk scales as 1/S.

Disproves the conjecture that risk scales as log S / S.

Establishes fundamental limits for finite-memory Bernoulli estimation.

Abstract

In this paper we consider the problem of estimating a Bernoulli parameter using finite memory. Let $X_1,X_2,\ldots$ be a sequence of independent identically distributed Bernoulli random variables with expectation $\theta$, where $\theta \in [0,1]$. Consider a finite-memory deterministic machine with $S$ states, that updates its state $M_n \in \{1,2,\ldots,S\}$ 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 the machine outputs an estimate at each time point according to some fixed mapping from the state space to the unit interval. The quality of the estimation procedure is measured by the asymptotic risk, which is the long-term average of the instantaneous quadratic risk. The main contribution of this paper is an upper bound on the smallest worst-case asymptotic risk any such machine can attain. This bound coincides with a lower bound derived by Leighton and Rivest, to imply that $\Theta(1/S)$ is the minimax asymptotic risk for deterministic $S$-state machines. In particular, our result disproves a longstanding $\Theta(\log S/S)$ conjecture for this quantity, also posed by Leighton and Rivest.