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.

翻译：我们重新审视混合方法（亦称拉普拉斯方法），以研究一般指数族中的集中现象。结合该族对数配分函数对应的Bregman散度性质与超鞅的混合方法，我们建立了一个控制族参数与其有限样本参数估计量之间Bregman散度的通用界。该界具有时间一致性，并引入了一个将经典信息增益扩展至指数族的量，我们称之为Bregman信息增益。面向实际应用者，针对多个经典指数族（如高斯、伯努利、指数、威布尔、帕累托、泊松与卡方分布），我们具体化该新界，得到置信集与Bregman信息增益的显式形式。进一步，我们将所得置信界与当前最先进的时间一致性集中方法进行数值比较，表明该新方法具有竞争性效果。最后，通过若干示例应用，我们展示了所提集中界的实用优势。