Universality of Bayesian mixture predictors

TL;DR

This paper demonstrates that for any set of probability measures generating sequences, a Bayesian mixture of countably many measures can achieve the best possible asymptotic prediction performance, extending previous results.

Contribution

It proves the universality of Bayesian mixture predictors for arbitrary sets of measures in sequential probability forecasting.

Findings

Bayesian mixtures can attain minimax asymptotic performance.

The result extends to non-zero asymptotic error cases.

Contrasts with prior work showing suboptimality of Bayesian mixtures in non-realizable settings.

Abstract

The problem is that of sequential probability forecasting for finite-valued time series. The data is generated by an unknown probability distribution over the space of all one-way infinite sequences. It is known that this measure belongs to a given set C, but the latter is completely arbitrary (uncountably infinite, without any structure given). The performance is measured with asymptotic average log loss. In this work it is shown that the minimax asymptotic performance is always attainable, and it is attained by a convex combination of a countably many measures from the set C (a Bayesian mixture). This was previously only known for the case when the best achievable asymptotic error is 0. This also contrasts previous results that show that in the non-realizable case all Bayesian mixtures may be suboptimal, while there is a predictor that achieves the optimal performance.