Finite-time optimality of Bayesian predictors
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The problem of sequential probability forecasting is considered in the most general setting: a model set C is given, and it is required to predict as well as possible if any of the measures (environments) in C is chosen to generate the data. No assumptions whatsoever are made on the model class C, in particular, no independence or mixing assumptions; C may not be measurable; there may be no predictor whose loss is sublinear, etc. It is shown that the cumulative loss of any possible predictor can be matched by that of a Bayesian predictor whose prior is discrete and is concentrated on C, up to an additive term of order n, where n is the time step. The bound holds for every n and every measure in C. This is the first non-asymptotic result of this kind. In addition, a non-matching lower bound is established: it goes to infinity with n but may do so arbitrarily slow.
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