# Estimating a probability using finite memory

## Abstract

Let {*X*_{i} _{i=1} ^{∞} } be a sequence of independent Bernoulli random variables with probability *p* that *X*_{ i }=1 and probability *q*=1 − *p* that *X*_{ i }=0 for all *i*≥1. We consider time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter *p* which take *X*_{1}, ... as an input sequence. In particular, we describe an *n*-state deterministic estimation procedure that can estimate *p* with mean-square error *O*(log *n*/*n*) and an *n*-state probabilistic estimation procedure that can estimate *p* with mean-square error *O*(1/*n*). We prove that the *O*(1/*n*) bound is optimal to within a constant factor. In addition, we show that linear estimation procedures are just as powerful (up to the measure of mean-square error) as arbitrary estimation procedures. The proofs are based on the Markov Chain Tree Theorem.

## Keywords

IEEE Transaction Estimation Procedure Learn Automaton Underlying Graph Deterministic Automaton## Preview

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