概率中的信息：有限de Finetti定理的另一个信息论证明
Information in probability: Another information-theoretic proof of a finite de Finetti theorem
摘要：
我们回顾了一些历史的信息理论的方法，以获得核心结果的概率论，并指出部分最近在这方面的兴趣复苏，目前的进展沿着几个有趣的方向。然后给出有限值随机变量的de Finetti经典表示定理的一个有限版本的信息论证明。我们推导出一个上界的分布之间的相对熵的第一个$k$的序列中的$n$可交换的随机变量，和一个适当的混合产品分布。混合措施的特点是法律的经验措施的原始序列，和德Finetti的结果是恢复作为一个推论。这个证明很好地受到了与统计力学有关的吉布斯条件原理的启发，并且遵循了一系列吸引人的步骤。这些步骤所需的技术估计是通过使用信息论中称为“类型法”的组合工具来获得的。'
英文摘要：
We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mixing measure is characterised as the law of the empirical measure of the original sequence, and de Finetti's result is recovered as a corollary. The proof is nicely motivated by the Gibbs conditioning principle in connection with statistical mechanics, and it follows along an appealing sequence of steps. The technical estimates required for these steps are obtained via the use of a collection of combinatorial tools known within information theory as `the method of types.'
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