Bayes, E-values and Testing

Abstract: This paper studies relationships between Kolmogorov complexity, Shannon entropy, Bayes factors, E-values, and exchangeability testing. The focus is on negative log marginal or predictive probabilities -- what I.J.~Good termed the ``weight of evidence'' -- as a common evidence statistic linking coding, prediction, and sequential testing. The paper reviews the relevant information-theoretic and martingale tools, and discusses exchangeability testing via conformal e-prediction.