A Note on New Bernstein-type Inequalities for the Log-likelihood Function of Bernoulli Variables

TL;DR

This paper introduces a new Bernstein-type inequality for the log-likelihood of Bernoulli variables that remains stable regardless of parameter values, enhancing theoretical guarantees in likelihood-based methods.

Contribution

The paper presents a novel Bernstein-type inequality that is independent of Bernoulli parameters, improving theoretical bounds for likelihood-based analyses.

Findings

The new inequality is parameter-independent and more robust.

It strengthens theoretical results in community detection.

Applicable to various likelihood-based methods for binary data.

Abstract

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.