Pseudo log-likelihood is a type of maximum likelihood estimation (MLE) method used in various fields including contextual bandits, influence maximization of social networks, and causal bandits. However, in previous literature \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}, the log-likelihood function may not be bounded, which may result in the algorithm they proposed not well-defined. In this paper, we give a counterexample that the maximum pseudo log-likelihood estimation fails and then provide a solution to correct the algorithms in \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}.

翻译：伪对数似然是一种最大似然估计方法，广泛应用于上下文强盗、社交网络影响力最大化以及因果强盗等领域。然而，在已有文献\citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}中，对数似然函数可能无界，从而导致所提出的算法可能定义不良。本文给出一个反例，表明最大伪对数似然估计失效，并提出一个解决方案，以修正\citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}中的算法。