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A problem posed by Freund is how to efficiently track a small pool of expertsout of a much larger set. This problem was solved when Bousquet and Warmuthintroduced their mixing past posteriors (MPP) algorithm in 2001.  In Freund's problem the experts would normally be considered black boxes.However, in this paper we re-examine Freund's problem in case the experts haveinternal structure that enables them to learn. In this case the problem has twopossible interpretations: should the experts learn from all data or only fromthe subsequence on which they are being tracked? The MPP algorithm solves thefirst case. Our contribution is to generalise MPP to address the second option.The results we obtain apply to any expert structure that can be formalisedusing (expert) hidden Markov models. Curiously enough, for our interpretationthere are \emph{two} natural reference schemes: freezing and sleeping. For eachscheme, we provide an efficient prediction strategy and prove the relevant lossbound.

Freezing and Sleeping: Tracking Experts that Learn by Evolving Past Posteriors