伯努利定理的一个贝叶斯证明
A Bayesian Proof of the Bernoulli Theorem
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中文总结 AI 辅助
该研究给出伯努利定理的新贝叶斯证明，基于信息论思想提出刻画伯努利过程上确界的新泛函，并将高斯过程的相关分布现象推广至伯努利过程。
中文摘要 AI 辅助
我们给出了伯努利定理的一个新证明，该定理由Talagrand提出猜想，并由Bednorz和Latała的开创性工作证明。我们的方法基于信息论思想：将伯努利过程上确界的下界转化为柯西加性信道中贝叶斯估计的基本极限。这引出了一个新的信息论泛函，它刻画了伯努利过程的上确界，其作用类似于高斯过程的Fernique majorizing-measure泛函。同样的视角产生了分布上的强化结果：对于任意给定的指标分布，我们刻画了该指标与伯努利过程所有耦合中可达到的最大期望值。这将Fernique和Talagrand此前通过研究为高斯过程所揭示的一种现象推广到了伯努利过程。
英文摘要
We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Latała. 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.
