Abstract
Let {X i āi=1 } be a sequence of independent Bernoulli random variables with probability p that X i =1 and probability q=1 ā p that X i =0 for all iā„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.
This research was supported by the Bantrell Foundation and by NSF grant MCS-8006938.
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Abstract, E. (1983). Estimating a probability using finite memory. In: Karpinski, M. (eds) Foundations of Computation Theory. FCT 1983. Lecture Notes in Computer Science, vol 158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12689-9_109
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DOI: https://doi.org/10.1007/3-540-12689-9_109
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