论文标题
在非方向的图形模型中的对数符号密度估计
Log-concave density estimation in undirected graphical models
论文作者
论文摘要
我们研究了对数符号的密度估计的最大似然估计的问题，并位于对应于给定无向图$ g $的图形模型中。我们表明，最大似然估计（MLE）是几个帐篷函数的指数的产物，每个最大集团$ g $。虽然图形模型中的一组对数符合密度是无限尺寸的，但我们的结果表明，可以通过求解有限维凸优化问题来找到MLE。我们提供实施和一些示例。此外，我们表明MLE存在，并且只要样品数量大于$ g $ chordal时最大的$ g $集团的大小，就具有概率1。我们表明，当图$ g $是集团的不相交联合时，MLE是一致的。最后，我们讨论了$ g $的图形模型中的对数凸线密度在$ g $中具有对数 - concove分解的条件。
We study the problem of maximum likelihood estimation of densities that are log-concave and lie in the graphical model corresponding to a given undirected graph $G$. We show that the maximum likelihood estimate (MLE) is the product of the exponentials of several tent functions, one for each maximal clique of $G$. While the set of log-concave densities in a graphical model is infinite-dimensional, our results imply that the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation and a few examples. Furthermore, we show that the MLE exists and is unique with probability 1 as long as the number of sample points is larger than the size of the largest clique of $G$ when $G$ is chordal. We show that the MLE is consistent when the graph $G$ is a disjoint union of cliques. Finally, we discuss the conditions under which a log-concave density in the graphical model of $G$ has a log-concave factorization according to $G$.
