Concentration and Confidence for Discrete Bayesian Sequence Predictors
This work addresses uncertainty quantification for Bayesian sequence predictors, which is incremental as it extends known expected error bounds to high-probability results.
The paper tackles the problem of understanding the distribution of cumulative error in Bayesian sequence prediction, proving tight high-probability bounds on the error measured via Kullback-Leibler divergence and constructing confidence bounds for KL and Hellinger errors, with application to the Knows What It Knows framework achieving state-of-the-art bounds.
Bayesian sequence prediction is a simple technique for predicting future symbols sampled from an unknown measure on infinite sequences over a countable alphabet. While strong bounds on the expected cumulative error are known, there are only limited results on the distribution of this error. We prove tight high-probability bounds on the cumulative error, which is measured in terms of the Kullback-Leibler (KL) divergence. We also consider the problem of constructing upper confidence bounds on the KL and Hellinger errors similar to those constructed from Hoeffding-like bounds in the i.i.d. case. The new results are applied to show that Bayesian sequence prediction can be used in the Knows What It Knows (KWIK) framework with bounds that match the state-of-the-art.