Characterizing predictable classes of processes
Daniil Ryabko
In: UAI 2009, June 18-21, 2009, Montreal, Canada.

## 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 class $\C$ of stochastic processes. We are interested in predictors $\rho$ whose conditional probabilities converge to the true'' $\mu$-conditional probabilities if any $\mu\in\C$ is chosen to generate the data. We show that if such a predictor exists, then a predictor can also be obtained as a convex combination of a countably many elements of $\C$. In other words, it can be obtained as a Bayesian predictor whose prior is concentrated on a countable set. This result is established for two very different measures of performance of prediction, one of which is very strong, namely, total variation, and the other is very weak, namely, prediction in expected average Kullback-Leibler divergence.

EPrint Type: Conference or Workshop Item (Paper) Project Keyword UNSPECIFIED Computational, Information-Theoretic Learning with StatisticsLearning/Statistics & OptimisationTheory & Algorithms 5979 Daniil Ryabko 08 March 2010