A Bayesian Proof of the Bernoulli Theorem

We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Lata{\l}a. Our approach is based on information-theoretic ideas: lower bounds on the supremum of a Bernoulli process are translated to the fundamental limits of Bayesian estimation in a Cauchy additive channel. This leads to a new information-theoretic functional that characterizes Bernoulli-process suprema and plays a role analogous to Fernique's majorizing-measure functional for Gaussian processes. The same viewpoint yields a distributional strengthening: for any prescribed law of the index, we characterize the largest expected value attainable over all couplings of that index with the Bernoulli process. This extends to Bernoulli processes a phenomenon previously understood for Gaussian processes through the work of Fernique and Talagrand.