Universal filtering of individual sequences corrupted by noise

This paper addresses the problem of estimating the current or the next bit of an arbitrary individual binary sequence in the presence of i.i.d. noise. We extend the work of Feder et al. (1992) to the case of noisy observations. It is proved that a finite-memory (FM) machine can achieve the same performance as the best finite-state machine (FSM). It is also shown that there exists a sequential algorithm that attains the same performance as the optimal FM machine and hence also the best FSM.