  PAC-Bayesian analysis is a frequentist framework for incorporating prior
knowledge into learning. It was inspired by Bayesian learning, which allows
sequential data processing and naturally turns posteriors from one processing
step into priors for the next. However, despite two and a half decades of
research, the ability to update priors sequentially without losing confidence
information along the way remained elusive for PAC-Bayes. While PAC-Bayes
allows construction of data-informed priors, the final confidence intervals
depend only on the number of points that were not used for the construction of
the prior, whereas confidence information in the prior, which is related to the
number of points used to construct the prior, is lost. This limits the
possibility and benefit of sequential prior updates, because the final bounds
depend only on the size of the final batch.
  We present a novel and, in retrospect, surprisingly simple and powerful
PAC-Bayesian procedure that allows sequential prior updates with no information
loss. The procedure is based on a novel decomposition of the expected loss of
randomized classifiers. The decomposition rewrites the loss of the posterior as
an excess loss relative to a downscaled loss of the prior plus the downscaled
loss of the prior, which is bounded recursively. As a side result, we also
present a generalization of the split-kl and PAC-Bayes-split-kl inequalities to
discrete random variables, which we use for bounding the excess losses, and
which can be of independent interest. In empirical evaluation the new procedure
significantly outperforms state-of-the-art.
