We prove a new Bernstein-type inequality for the log-likelihood function of Bernoulli variables. In contrast to classical Bernstein's inequality and Hoeffding's inequality when applied to the log-likelihood, the new bound is independent of the parameters of the Bernoulli variables and therefore does not blow up as the parameters approach 0 or 1. The new inequality strengthens certain theoretical results on likelihood-based methods for community detection in networks and can be applied to other likelihood-based methods for binary data.

中文翻译：

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关于伯努利变量对数似然函数的新伯恩斯坦型不等式的注解

我们证明了伯努利变量的对数似然函数的新伯恩斯坦型不等式。与应用于对数似然的经典 Bernstein 不等式和 Hoeffding 不等式相比，新的界限与伯努利变量的参数无关，因此不会随着参数接近 0 或 1 而爆炸。新的不等式加强了某些理论结果用于网络中社区检测的基于似然的方法，并且可以应用于其他基于似然的二进制数据方法。