Characterizing predictable classes of processes

TL;DR

This paper investigates the conditions under which sequence prediction is possible for classes of stochastic processes, showing that Bayesian predictors based on countable mixtures can achieve convergence in various divergence measures.

Contribution

It proves that if a predictor exists for a class of processes, then a Bayesian predictor with a countable prior can also perform well, for both strong and weak convergence measures.

Findings

Predictors can be constructed as countable mixtures of processes in the class.

Convergence of predictors is established for total variation and expected average Kullback-Leibler divergence.

Bayesian predictors are sufficient for universal prediction within the class.

Abstract

The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ 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.