AI 中文总结
本文通过信息恒等式将偏差事件概率指数分解为速率与依赖性两项，揭示其几何结构，并推广至一般参考律，给出指数测度变化期望的精确值。
AI 中文摘要
在偏差事件中，$n$ 次独立抽取的经验测度落在一组分布集合中。其概率的指数在每一个 $n$ 处分裂为两个非负项。第一项是速率：$n$ 乘以从总体到事件下典型抽取分布的相对熵。第二项是条件作用在抽取之间引入的依赖性，通过它们的总相关来度量。它们的相对大小取决于集合的几何结构。本文关注这一精确分裂及其两项相互作用所揭示的几何结构。当事件将每次抽取限制在单一集合中时，依赖性恰好消失，且该约束的经典速率是精确的。在一个紧致群作用下不变的集合上，若该群固定总体且不留下任何中间概率的不变集合，则速率消失；该群置换的假设类上的均匀收敛为1。在一个阈值位于均值之上固定标准误差数的半空间上，指数中属于依赖性的比例趋向于仅由事件概率决定的函数。在一个被分割成若干部分的集合上，依赖性等于各部分的平均值加上 $n$ 乘以它们边缘分布的 Jensen-Shannon 散度，再减去它们权重的熵。该分裂可推广到一般参考律。当信息投影是这种律时，该分裂给出了到投影的指数测度变化中期望的精确值。
英文摘要
In a deviation event, the empirical measure of $n$ independent draws lands in a set of distributions. The exponent of its probability splits, at every $n$, into two nonnegative terms. The first is the rate: $n$ times the relative entropy, from the population, of the distribution of a typical draw under the event. The second is the dependence that conditioning induces among the draws, measured by their total correlation. Their relative sizes depend on the geometry of the set. This paper focuses on this exact split and on the geometry that the interplay of its two terms reveals. The dependence vanishes exactly when the event confines every draw to a single set, and the classical rate of that constraint is exact. On a set invariant under a compact group that fixes the population and leaves no invariant set of intermediate probability, the rate vanishes; uniform convergence over a hypothesis class that such a group permutes is one. On a half-space whose threshold sits a fixed number of standard errors above the mean, the fraction of the exponent that is dependence tends to a function of the event's probability alone. On a set split into pieces, the dependence is the pieces' average plus $n$ times the Jensen-Shannon divergence of their marginals, minus the entropy of their weights. The split extends to a general reference law. When the information projection is such a law, the split gives the exact value of the expectation in the exponential change of measure to the projection.
