On Finding Predictors for Arbitrary Families of Processes

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 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.