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### PAC-Bayesian Inequalities for Martingales

We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. Our results extend the PAC-Bayesian analysis in learning theory from the i.i.d. setting to martingales opening the way for its application to importance weighted sampling, reinforcement learning, and other interactive learning domains, as well as many other domains in probability theory and statistics, where martingales are encountered. We also present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0,1] interval by the expectation of the same function of independent Bernoulli variables. This inequality is applied to derive a tighter analog of Hoeffding-Azuma's inequality.

### Testing Exchangeability On-Line

The majority of theoretical work in machine learning is done under the assumption of exchangeability: essentially, it is assumed that the examples are generated from the same probability distribution independently. This paper is concerned with the problem of testing the exchangeability assumption in the online mode: examples are observed one by one and the goal is to monitor online the strength of evidence against the hypothesis of exchangeability. We introduce the notion of exchangeability martingales, which are online procedures for detecting deviations from exchangeability; in essence, they are betting schemes that never risk bankruptcy and are fair under the hypothesis of exchangeability. Some specific exchangeability martingales are constructed using Transductive Confidence Machine.