一般指数族的 Bregman 偏差
Bregman Deviations of Generic Exponential Families
AI总结：
本文利用 Laplace 混合法与 Bregman 散度建立一般指数族参数估计的时间一致集中界，并在多类分布和应用中验证其竞争力。
摘要：
我们重新审视混合方法技术（也称为 Laplace 方法），以研究一般指数族中的集中现象。将与该族的对数配分函数相关的 Bregman 散度性质与用于超鞅的混合方法相结合，我们建立了一个通用界，用于控制该族参数与该参数的有限样本估计量之间的 Bregman 散度。我们的界是时间一致的，并产生了一个将经典信息增益推广到指数族的量，我们称之为 Bregman 信息增益。对于实践者，我们将这一新界实例化到若干经典族，例如 Gaussian、Bernoulli、Exponential、Weibull、Pareto、Poisson 和 Chi-square，从而得到置信集和 Bregman 信息增益的显式形式。我们进一步在数值上将所得置信界与时间一致集中的最先进替代方法进行比较，并表明这种新方法产生了具有竞争力的结果。最后，我们在一些示例性应用中强调了我们的集中界的优势。
英文摘要：
We revisit the method of mixture technique, also known as the Laplace method, to study the concentration phenomenon in generic exponential families. Combining the properties of Bregman divergence associated with log-partition function of the family with the method of mixtures for super-martingales, we establish a generic bound controlling the Bregman divergence between the parameter of the family and a finite sample estimate of the parameter. Our bound is time-uniform and makes appear a quantity extending the classical information gain to exponential families, which we call the Bregman information gain. For the practitioner, we instantiate this novel bound to several classical families, e.g., Gaussian, Bernoulli, Exponential, Weibull, Pareto, Poisson and Chi-square yielding explicit forms of the confidence sets and the Bregman information gain. We further numerically compare the resulting confidence bounds to state-of-the-art alternatives for time-uniform concentration and show that this novel method yields competitive results. Finally, we highlight the benefit of our concentration bounds on some illustrative applications.
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