Estimating a probability using finite memory

Let { X, }z 1 be a sequence of independent Bernoulli random variables with probability p that Xi = 1 and probability 4 = 1 p that X, = 0 for all i 2 1. Time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter p are considered which take X,, . as an input sequence. In particular, an n-state deterministic estimation procedure is described which can estimate p with mean-square error O(logn/n) and an n-state probabilistic estimation procedure which can estimate p with mean-square error 0(1/n). It is proved that the 0(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.