Estimating a probability using finite memory

Let\{Xᵢ\}ᵢ₌₁∞be a sequence of independent Bernoulli random variables with probabilitypthatXᵢ = 1and probabilityq=1-pthatXᵢ = 0for alli ≥ 1. Time-invariant finite-memory (i.e., finite-state) estimation procedures for the parameter p are considered which takeX₁, an input sequence. In particular, an n-state deterministic estimation procedure is described which can estimate p with mean-square errorO(log n/n)and ann-state probabilistic estimation procedure which can estimatepwith mean-square errorO(1/n). It is proved that theO(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.