We propose simple randomized strategies for sequential prediction under
imperfect monitoring, that is, when the forecaster does not have access to the
past outcomes but rather to a feedback signal. The proposed strategies are
consistent in the sense that they achieve, asymptotically, the best possible
average reward. It was Rustichini (1999) who first proved the existence of such
consistent predictors. The forecasters presented here offer the first
constructive proof of consistency. Moreover, the proposed algorithms are
computationally efficient. We also establish upper bounds for the rates of
convergence. In the case of deterministic feedback, these rates are optimal up
to logarithmic terms.Comment: Journal version of a COLT conference pape