AI 中文总结
本文提出归一化期望最大化（N-EM）算法，用于一类带积分的对数似然函数。该算法继承EM算法优点，由归一化、期望、最大化三步组成，能解决EM算法无法处理的问题，还可统一应用于EM能处理的问题，通过数值实验建立了收敛性质。
AI 中文摘要
尽管期望最大化（EM）算法是统计学中强大的优化工具，但仅适用于缺失/不完整数据问题或具有潜在变量结构的问题。引入潜在变量是一种技巧。本文提出了归一化EM（N-EM）算法用于一类带积分的对数似然函数。它作为原始EM算法的扩展，继承其优点，由归一化步骤（构建归一化密度函数）、期望步骤（计算代理Q函数）和最大化步骤组成。探讨了上升性质、归一化密度函数的最佳选择等。通过实际应用表明，N-EM算法能解决EM算法无法处理的问题，且可统一应用于EM能处理的问题。还进行了数值实验并建立了收敛性质。
英文摘要
Although the $\textit{expectation-maximization}$ (EM) algorithm is a powerful optimization tool in statistics, it can only be applied to missing/incomplete data problems or to problems with a latent-variable structure. It is well known that the introduction of latent variables (or the data augmentation) is an art; i.e., it could only be done case by case. In this paper, we propose a new algorithm, a so-called $\textit{normalized EM}$ (N-EM) algorithm, for a class of log-likelihood functions with integrals. As an extension of the original EM algorithm, the N-EM algorithm inherits all advantages of EM-type algorithms and consists of three steps: normalization step (N-step), expectation step (E-step) and maximization step (M-step), where the N-step is to construct a $\textit{normalized density function}$ (ndf), the E-step is to compute a well-established surrogate $Q$-function and the M-step is to maximize the $Q$-function as in the original EM algorithm. The ascent property, the best choice of the ndf, and those N-EM algorithms with a difficult M-step are also explored. By multiple real applications, we have shown that the N-EM algorithm can solve some problems which cannot be addressed by the EM algorithm. Next, for problems to which the EM can be applied (often case by case), the N-EM algorithm can be employed in a unified framework. Numerical experiments are performed and convergence properties are also established.
