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. Time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter p are considered which take X 1, … as an input sequence. In particular, an n-state deterministic estimation procedure is described which can estimate p with mean-square error O(log n/n) and an n-state probabilistic estimation procedure which can estimate p with mean-square error O(1/n). It is proved that the O(1/n) bound is optimal to within a constant factor. In addition, it is shown 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 an analog of the well-known matrix tree theorem that is called the Markov chain tree theorem. Copyright © 1986 by The Institute of Electrical and Electronics Engineers, Inc.