Zhou Hua, Lange Kenneth
Post-Doctoral Fellow, Department of Human Genetics, University of California, Los Angeles, CA 90095-7088 (
J Comput Graph Stat. 2010 Sep 1;19(3):645-665. doi: 10.1198/jcgs.2010.09014.
The MM (minorization-maximization) principle is a versatile tool for constructing optimization algorithms. Every EM algorithm is an MM algorithm but not vice versa. This article derives MM algorithms for maximum likelihood estimation with discrete multivariate distributions such as the Dirichlet-multinomial and Connor-Mosimann distributions, the Neerchal-Morel distribution, the negative-multinomial distribution, certain distributions on partitions, and zero-truncated and zero-inflated distributions. These MM algorithms increase the likelihood at each iteration and reliably converge to the maximum from well-chosen initial values. Because they involve no matrix inversion, the algorithms are especially pertinent to high-dimensional problems. To illustrate the performance of the MM algorithms, we compare them to Newton's method on data used to classify handwritten digits.
MM（最小化-最大化）原理是构建优化算法的一种通用工具。每个期望最大化（EM）算法都是一个MM算法，但反之则不成立。本文推导了用于离散多元分布（如狄利克雷多项分布和康纳-莫西曼分布、内尔查尔-莫雷尔分布、负多项分布、某些划分上的分布以及零截断和零膨胀分布）的最大似然估计的MM算法。这些MM算法在每次迭代时都会增加似然性，并从精心选择的初始值可靠地收敛到最大值。由于它们不涉及矩阵求逆，因此这些算法特别适用于高维问题。为了说明MM算法的性能，我们将它们与牛顿法在用于手写数字分类的数据上进行比较。
