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Combinatorial EM algorithms

机译：组合EM算法

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The complete-data model that underlies an Expectation-Maximization (EM) algorithm must have a parameter space that coincides with the parameter space of the observed-data model. Otherwise, maximization of the observed-data log-likelihood will be carried out over a space that does not coincide with the desired parameter space. In some contexts, however, a natural complete-data model may be defined only for parameter values within a subset of the observed-data parameter space. In this paper we discuss situations where this can still be useful if the complete-data model can be viewed as a member of a finite family of complete-data models that have parameter spaces which collectively cover the observed-data parameter space. Such a family of complete-data models defines a family of EM algorithms which together lead to a finite collection of constrained maxima of the observed-data log-likelihood. Maximization of the log-likelihood function over the full parameter space then involves identifying the constrained maximum that achieves the greatest log-likelihood value. Since optimization over a finite collection of candidates is referred to as combinatorial optimization, we refer to such a family of EM algorithms as a combinatorial EM (CEM) algorithm. As well as discussing the theoretical concepts behind CEM algorithms, we discuss strategies for improving the computational efficiency when the number of complete-data models is large. Various applications of CEM algorithms are also discussed, ranging from simple examples that illustrate the concepts, to more substantive examples that demonstrate the usefulness of CEM algorithms in practice.
机译：期望最大化（EM）算法基础的完整数据模型必须具有与观察数据模型的参数空间一致的参数空间。否则，将在与所需参数空间不一致的空间上执行观察数据对数似然性的最大化。但是，在某些情况下，可以仅为观察数据参数空间的子集中的参数值定义自然的完整数据模型。在本文中，我们讨论了以下情况：如果可以将完整数据模型视为具有参数空间的有限系列完整数据模型的成员，那么这些参数空间将共同覆盖观察数据参数空间。这样的一组完整数据模型定义了一系列EM算法，这些算法共同导致了观测数据对数似然的约束最大值的有限集合。然后，在整个参数空间上最大化对数似然函数包括确定达到最大对数似然值的约束最大值。由于对有限候选集合的优化称为组合优化，因此我们将此类EM算法家族称为组合EM（CEM）算法。除了讨论CEM算法背后的理论概念外，我们还将讨论在完整数据模型数量很大时提高计算效率的策略。还讨论了CEM算法的各种应用，从说明概念的简单示例到展示CEM算法在实践中有用的更实质性的示例。

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