On Finding Predictors for Arbitrary Families of Processes
Daniil Ryabko
Journal of Machine Learning Research Volume 11, Number Feb, pp. 581-602, 2010. ISSN 1532-4435

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Abstract

The problem is sequence prediction in the following setting. A sequence $x_1,\dots,x_n,\dots$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $\mu$. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure $\mu$ belongs to an arbitrary but known class $\C$ of stochastic process measures. We are interested in predictors $\rho$ whose conditional probabilities converge (in some sense) to the true'' $\mu$-conditional probabilities, if any $\mu\in\C$ is chosen to generate the sequence. The contribution of this work is in characterizing the families $\C$ for which such predictors exist, and in providing a specific and simple form in which to look for a solution. We show that if any predictor works, then there exists a Bayesian predictor, whose prior is discrete, and which works too. We also find several sufficient and necessary conditions for the existence of a predictor, in terms of topological characterizations of the family $\C$, as well as in terms of local behaviour of the measures in $\C$, which in some cases lead to procedures for constructing such predictors. It should be emphasized that the framework is completely general: the stochastic processes considered are not required to be i.i.d., stationary, or to belong to any parametric or countable family.

EPrint Type: Article Project Keyword UNSPECIFIED Computational, Information-Theoretic Learning with StatisticsLearning/Statistics & OptimisationTheory & Algorithms 5966 Daniil Ryabko 08 March 2010