Estimating a probability using finite memory

Abstract

Let {Xii=1∞} be a sequence of independent Bernoulli random variables with probability p that Xi=1 and probability q=1 − p that Xi=0 for all i≥1. We consider time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter p which take X1,.. 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.